Analytic Number Theory

Analytic Number Theory Developments in Mathematics VOLUME 6 Edited by Series Editor: Chaohua Jia Krishnaswami Allad...

Author: Chaohua Jia | Kohji Matsumoto

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Analytic Number Theory

Developments in Mathematics VOLUME 6

Edited by

Series Editor:

Chaohua Jia

Krishnaswami Alladi, University of Florida, U.S.A.VOLUME3

Academia Sinica, China

Series Editor:

and

Krishnaswami Alladi, University of Florida, U.S.A.

Kohji Matsumoto Nagoya University, Japan

Aims and Scope Developments in Mathematics is a book series publishing

(i) Proceedings of Conferences dealing with the latest research advances, (ii) Research Monographs, and (iii) Contributed Volumes focussing on certain areas of special interest. Editors of conference proceedings are urged to include a few survey papers for wider appeal. Research monographs which could be used as texts or references for graduate level courses would also be suitable for the series. Contributed volumes are those where various authors either write papers or chapters in an organized volume devoted to a topic of speciaYcurrent interest or importance. A contributed volume could deal with a classical topic which is once again in the limelight owing to new developments.

I

KLUWER ACADEMIC PUBLISHERS DORDRECHT I BOSTON I LONDON

A C.I.P. Catalogue record for this book is available from the Library of Congress.

Contents

ISBN 1-4020-0545-8

Preface Published by Kluwer Academic Publishers, P.O. Box 17,3300AA Dordrecht, The Netherlands. Sold and distributed in North, Central and South America by Kluwer Academic Publishers, 101 Philip Drive, Norwell, MA 02061, U.S.A. In all other countries, sold and distributed by Kluwer Academic Publishers, PO. Box 322,3300 AH Dordrecht, The Netherlands.

1 On analytic continuation of multiple L-functions and related zetafunctions

Shzgekz AKIYAMA, Hideakz ISHIKA WA L

On the values of certain q-hypergeometric series I1

Masaaki A MOU, Masanori KATSURADA, Keijo V AANA NEN 3

The class number one problem for some non-normal sextic CMfields Gdrard BO UTTEAUX, Stiphane L 0 UBO UTIN 4 Ternary problems in additive prime number theory

Jog BRUDERN, Koichi KA WADA Printed on acid-free paper

5

A generalization of E. Lehmer 's congruence and its applications Tiamin CAI 6 On Chen's theorem CAI Yingchun, LU Minggao All Rights Reserved O 2002 Kluwer Academic Publishers No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner Printed in the Netherlands.

7 On a twisted power mean of L(1, X )

Shigekz EGAMI 8 On the pair correlation of the zeros of the Riemann zeta function A k a FUJI1

vi

Contents

ANALYTIC NUMBER THEORY

9

21 On families of cubic Thue equations Isao WAKA BAYASHI

Discrepancy of some special sequences Kazuo GOTO, Yubo OHKUBO 10 Pad6 approximation to the logarithmic derivative of the Gauss hypergeometric function Masayoshi HATA, Marc HUTTNER 11 The evaluation of the sum over arithmetic progressions for the coefficients of the Rankin-Selberg series I1 Yumiko ICHIHARA

157

23 173

12 Substitutions, atomic surfaces, and periodic beta expansions Shunji ITO, Yuki SANO 13 The largest prime factor of integers in the short interval Chaohua JIA 14 A general divisor problem in Landau's framework S. KANEMITSU, A. SANKARANARAYANAN

15 On inhomogeneous Diophantine approximation and the Borweins' algorithm, I1 Takao KOMATSU

223

16 Asymptotic expansions of double gamma-functions and related remarks Kohji MATSUMOTO

243

A note on a certain average of L ( $ Leo MURATA

+ it, X )

18 On covering equivalence Zhi- Wei SUN Certain words, tilings, their non-periodicity, and substitutions of high dimension Jun-ichi TA MURA 20 Determination of all Q-rational CM-points in moduli spaces of polarized abelian surfaces Atsuki UMEGAKI

Two examples of zeta-regularization Masanri YOSHIMOTO

303

349

A h brid mean value formula of Dedekind sums and Hurwitz zetaLnctions ZHANG Wenpeng

vii 359

Preface

From September 13 to 17 in 1999, the First China- Japan Seminar on Number Theory was held in Beijing, China, which was organized by the Institute of Mathematics, Academia Sinica jointly with Department of Mathematics, Peking University. Ten Japanese Professors and eighteen Chinese Professors attended this seminar. Professor Yuan Wang was the chairman, and Professor Chengbiao Pan was the vice-chairman. This seminar was planned and prepared by Professor Shigeru Kanemitsu and the first-named editor. Talks covered various research fields including analytic number theory, algebraic number theory, modular forms and transcendental number theory. The Great Wall and acrobatics impressed Japanese visitors. From November 29 to December 3 in 1999, an annual conference on analytic number theory was held in Kyoto, Japan, as one of the conferences supported by Research Institute of Mat hemat ical Sciences (RIMS), Kyoto University. The organizer was the second-named editor. About one hundred Japanese scholars and some foreign visitors corning from China, France, Germany and India attended this conference. Talks covered many branches in number theory. The scenery in Kyoto, Arashiyarna Mountain and Katsura River impressed foreign visitors. An informal report of this conference was published as the volume 1160 of Siirikaiseki Kenkyiisho Kakyiiroku (June 2000), published by RIMS, Kyoto University. The present book is the Proceedings of these two conferences, which records mainly some recent progress in number theory in China and Japan and reflects the academic exchanging between China and Japan. In China, the founder of modern number theory is Professor Lookeng Hua. His books "Introduction to Number Theory", "Additive Prime Number Theory" and so on have influenced not only younger generations in China but also number theorists in other countries. Professor Hua created the strong tradition of analytic number theory in China. Professor Jingrun Chen did excellent works on Goldbach's conjecture. The report literature of Mr. Chi Xu "Goldbach Conjecture" made many

x


people out of the circle of mathematicians to know something on number theory. In Japan, the first internationally important number theorist is Professor Teiji Takagi, one of the main contributors to class field theory. His books "Lectures on Elementary Number Theory" and "Algebraic Number Theory" (written in Japanese) are still very useful among Japanese number theorists. Under the influence of Professor Takagi, a large part of research of the first generation of Japanese analytic number theorists such as Professor Zyoiti Suetuna, Professor Tikao Tatuzawa and Professor Takayoshi Mitsui were devoted to analytic problems on algebraic number fields. Now mathematicians of younger generations have been growing in both countries. It is natural and necessary to exchange in a suitable scale between China and Japan which are near in location and similar in cultural background. In his visiting to Academia Sinica twice, Professor Kanemitsu put forward many good suggestions concerning this matter and pushed relevant activities. This is the initial driving force of the project of the First China-Japan Seminar. Here we would like to thank sincerely Japanese Science Promotion Society and National Science Foundation of China for their great support, Professor Yuan Wang for encouragement and calligraphy, Professor Yasutaka Ihara for his support which made the Kyoto Conference realizable, Professor Shigeru Kanemitsu and Professor Chengbiao Pan for their great effort of promot ion. Since many attendants of the China-Japan Seminar also attended the Kyoto Conference, we decided to make a plan of publishing the joint Proceedings of these two conferences. It was again Professor Kanemitsu who suggested the way of publishing the Proceedings as one volume of the series "Developments in Mathematics", Kluwer Academic Publishers, and made the first contact to Professor Krishnaswami Alladi, the series editor of this series. We greatly appreciate the support of Professor Alladi. We are also indebted to Kluwer for publishing this volume and to Mr. John Martindale and his assistant Ms. Angela Quilici for their constant help. These Proceedings include 23 papers, most of which were written by participants of at least one of the above conferences. Professor Akio Fujii, one of the invited speakers of the Kyoto Conference, could not attend the conference but contributed a paper. All papers were refereed. We since~elythank all the authors and the referees for their contributions. Thanks are also due to Dr. Masami Yoshimoto, Dr. Hiroshi Kumagai, Dr. Jun Furuya, Dr. Yumiko Ichihara, Mr. Hidehiko Mishou, Mr. Masatoshi Suzuki, and especially Dr. Yuichi Kamiya for their effort

PREFACE

xi

of making files of Kluwer LaTeX style. The contents include several survey or half-survey articles (on prime numbers, divisor problems and Diophantine equations) as well as research papers on various aspects of analytic number theory such as additive problems, Diophantine approximations and the theory of zeta and L-functions. We believe that the contents of the Proceedings reflect well the main body of mathematical activities of the two conferences. The Second China-Japan Seminar was held from March 12 to 16,2001, in Iizuka, Fukuoka Prefecture, Japan. The description of this conference will be found in the coming Proceedings. We hope that the prospects of the exchanging on number theory between China and Japan will be as beautiful as Sakura and plum blossom. April 2001 CHAOHUA JIA AND KOHJIMATSUMOTO (EDITORS)

...

Xlll

S. Akiyama T. Cai Y. Cai X. Cao Y. Chen S. Egami X. Gao M. Hata C. Jia

S. Kanemitsu H. Li C. Liu 3. Liu M. Lu K. Matsumoto 2. Meng K. Miyake L. Murata Y. Nakai

C. B. Pan 2. Sun

Y. Tanigawa I. Wakabayashi W. Wang Y. Wang G. Xu W. Zhai W. Zhang

xiv


LIST OF PARTICIPANTS (Kyoto) (This is only the list of participants who signed the sheet on the desk at the entrance of the lecture room.)

T . Adachi S. Akiyama M. Amou K. Azuhata J. Briidern K. Chinen S. Egami J. Furuya Y. Gon K. Got0 Y, Hamahata T. Harase M. Hata K. Hatada T. Hibino M. Hirabayashi Y. Ichihara Y. Ihara M. Ishibashi N. Ishii Hideaki Ishikawa Hirofumi Ishikawa S. It0 C. Jia T . Kagawa Y. Kamiya M. Kan S. Kanemitsu H. Kangetu T. Kano T. Kanoko N. Kataoka M. Katsurada

K. Kawada H. Kawai Y. Kitaoka I. Kiuchi Takao Komatsu Toru Komatsu Y. Koshiba Y. Koya S. Koyama H. Kumagai M. Kurihara T. Kuzumaki S. Louboutin K. Matsumoto H. Mikawa H. Mishou K. Miyake T. Mizuno R. Morikawa N. Murabayashi L. Murata K. Nagasaka M. Nagata H. Nagoshi D. Nakai Y. Nakai M. Nakajima K. Nakamula I. Nakashima M. Nakasuji K. Nishioka T. Noda J. Noguchi

Y. Ohkubo Y. Ohno T . Okano R. Okazaki Y. Okuyama Y. onishi T . P. Peneva K. Saito A. Sankaranarayanan Y. Sano H. Sasaki R. Sasaki I. Shiokawa M. Sudo T. Sugano M. Suzuki S. Suzuki I. Takada R. Takeuchi A. Tamagawa J . Tamura T . Tanaka Y. Tanigawa N. Terai T . Toshimitsu Y. Uchida A. Umegaki I. Wakabayas hi A. Yagi M. Yamabe S. Yasutomi M. Yoshimoto W. Zhang

O N ANALYTIC CONTINUATION OF MULTIPLE L-FUNCTIONS AND RELATED ZETA-FUNCTIONS Shigeki AKIYAMA Department of Mathematics, Faculty of Science, Niigata University, Ikarashi 2-8050, Niigata 950-2181, Japan [email protected]

Hideaki ISHIKAWA Graduate school of Natural Science, Niigata University, Ikarashi 2-8050, Niigata 9502181, Japan i~ikawah@~ed.sc.niigata-u.ac.jp

Keywords: Multiple L-function, Multiple Hurwitz zeta function, Euler-Maclaurin summation formula Abstract

A multiple L-function and a multiple Hurwitz zeta function of EulerZagier type are introduced. Analytic continuation of them as complex functions of several variables is established by an application of the Euler-Maclaurin summation formula. Moreover location of singularities of such zeta functions is studied in detail.

1991 Mathematics Subject Classification: Primary 1lM41; Secondary 32Dxx, 11MXX, llM35.

1.

INTRODUCTION

Analytic continuation of Euler-Zagier's multiple zeta function of two variables was first established by F. V. Atkinson [3] with an application t o the mean value problem of the Riemann zeta function. We can find recent developments in (81, [7] and [5]. From an analytic point of view, these results suggest broad applications of multiple zeta functions. In [9] and [lo], D. Zagier pointed out an interesting interplay between positive integer values and other areas of mathematics, which include knot theory and mathematical physics. Many works had been done according to his motivation but here we restrict our attention to the analytic contin-

2


On analytic continuation of multiple L-functions and related zeta-functions

uation. T . Arakawa and M. Kaneko [2] showed an analytic continuation with respect to the last variable. To speak about the analytic continuation with respect to all variables, we have to refer to J. Zhao [ll]and S. Akiyama, S. Egami and Y. Tanigawa [I]. In [ll],an analytic continuation and the residue calculation were done by using the theory of generalized functions in the sense of I. M. Gel'fand and G. E. Shilov. In [I],they gave an analytic continuation by means of a simple application of the Euler-Maclaurin formula. The advantage of this method is that it gives the complete location of singularities. This work also includes some study on the values at non positive integers. In this paper we consider a more general situation, which seems important for number theory, in light of the method of [I]. We shall give an analytic continuation of multiple Hurwitz zeta functions (Theorem 1) and also multiple L-functions (Theorem 2) defined below. In special cases, we can completely describe the whole set of singularities, by using a property of zeros of Bernoulli polynomials (Lemma 4) and a non vanishing result on a certain character sum (Lemma 2). We explain notations used in this paper. The set of rational integers is denoted by Z, the rational numbers by Q, the complex numbers by @ and the positive integers by N. We write Z

where ni E N (i = 1,..., k). If W(si) 1 (i = 1 , 2,... , k - 1) and W(sk) > 1, then these series are absolutely convergent and define holomorphic functions of k complex variables in this region. In the sequel we write them by (s I p ) and Lk( s I x), for abbreviation. The Hurwitz zeta function 3

are the possible singularities, a s desired. Note that the singularities of the form sk-2 + s k - l +sk + r = 1,-1,-3,-5 ,... may appear. However, these singularities don't affect our description. Next we will show that they are the 'real' singularities. For example, sk-l+ s k = 7 occurs in several ways the singularities of the form s k - 2 for a fixed 7. So our task is to show that no singularities defined by one of the above equations will identically vanish in the summation process. This can be shown by a small trick of replacing variables:

+

14

Proof of Corollary 1. Considering the case k = 2 in Theorem 2, we see

In fact, we see that the singularities of C k ( ~ l , - . . , u k - 2 , ~ k - Ul k r U k

15

On analytic continuation of multiple L-finctions and related zeta-functions

ANALYTICNUMBERTHEORY

IPl,.--,Pk)

appear in

+

By this expression we see that the singularities of (ul, . . . ,uk-1 r I PI, . . . ,8k-I) are summed with functions of uk of dzfferent degree. Thus these singularities, as weighted sum by another variable uk, will not vanish identically. This argument seems to be an advantage of [I], which clarify the exact location of singularities. The Theorem is proved 0 by the induction.

ANALYTIC CONTINUATION OF MULTIPLE L-FUNCTIONS Proof of Theorem 2. When %si > 1 for i = 1 , 2 , . . . , k,

We have a meromorphic continuation of L2(sI X) to C2, which is holomorphic in the domain (3). Note that the singularities occur in

4.

the series is

absolutely convergent. Rearranging the terms,

and

l o o

If ~2 is not principal then the first term vanishes and we see the 'singular part' is

Thus we get the result by using Lemma 2 and the fact: By this expression, it suffices to show that the series in the last brace has the desirable property. When ai - ai+l >_ 0 holds for z = 1,. . . ,k- 1, this is clear form Theorem 1, since this series is just a multiple Hurwitz zeta function. Proceeding along the same line with the proof of Theorem 1, other cases are also easily deduced by recursive applications of Lemma 1. Since there are no need to use binomial expansions, this case is easier than before. I7

for n 2 1 and a non principal character X.

0

AS we stated in the introduction, we do not have a satisfactory answer to the problem of describing whole sigularities of multiple L-functions

16

A N A L Y T I C N U M B E R THEORY

in the case k 2 3, at present. For example when k = 3, what we have to show is the non vanishing of the sum:

apart from trivial cases.

O N THE VALUES OF CERTAIN Q-HYPERGEOMETRIC SERIES I1

References [I] S. Akiyama, S. Egami, and Y. Tanigawa , An analytic continuation of multiple zeta functions and their values at non-positive integers, Acta Arith. 98 (2001), 107-116. [2] T . Arakawa and M. Kaneko, Multiple zeta values, poly-Bernoulli numbers, and related zeta functions, Nagoya Math. J. 153 (1999), 189-209. [3] F. V. Atkinson, The mean value of the Riemann zeta-function, Acta Math., 81 (1949), 353-376. [4] K. Dilcher, Zero of Bernoulli, generalized Bernoulli and Euler polynomials, Mem. Amer. Math. Soc., Number 386, 1988. [5] S. Egami, Introduction to multiple zeta function, Lecture Note at Niigata University (in Japanese). [6] K. Inkeri, The real roots of Bernoulli polynomials, Ann. Univ. Turku. Ser. A I37 (1959), 20pp. [7] M. Katsurada and K. Matsumoto, Asymptotic expansions of the mean values of Dirichlet L-functions. Math. Z., 208 (1991), 23-39. [8] Y. Motohashi, A note on the mean value of the zeta and L-functions. I, Proc. Japan Acad., Ser. A Math. Sci. 61 (1985), 222-224. [9] D. Zagier, Values of zeta functions and their applications, First European Congress of Mathematics, Vol. 11, Birkhauser, 1994, pp.497512. [lo] D. Zagier, Periods of modular forms, traces of Hecke operators, and multiple zeta values, Research into automorphic forms and L functions (in Japanese) (Kyoto, 1N Z ) , Siirikaisekikenkyusho K6kytiroku, 843 (1993), 162-170. [Ill .I. Zhao, Analytic continuation of multiple zeta functions, Proc. Amer. Math. Soc., 128 (2000), 1275-1283.

Masaaki AMOU Department of Mathematics, Gunma University, Tenjin-cho 1-5-1, Kiryu 376-8515, Japan [email protected]~i.~unma-u.ac.jp

Masanori KATSURADA Mathematics, Hiyoshi Campus, Keio University, Hiyoshi 4-1-1, hama 223-8521, Japan

Kohoku-ku, Yoko-

[email protected]

Keijo VAANANEN Department of Mathematics, University of Oulu, P. 0 . Box 3000, SF-90014 Finland

Oulu,

[email protected]

Keywords: Irrationality, Irrationality measure, q-hypergeometric series, q-Bessel function, S-unit equation

Abstract

As a continuation of the previous work by the authors having the same title, we study the arithmetical nature of the values of certain qhypergeometric series $ ( z ; q) with a rational or an imaginary quadratic integer q with (ql > 1, which is related to a q-analogue of the Bessel function Jo(z). The main result determines the pairs (q, a) with cr E K for which 4(a;q) belongs to K , where K is an imaginary quadratic number field including q.

2000 Mathematics Subject Classification. Primary: 11572; Secondary: 11582.

The first named author was supported in part by Grant-in-Aid for Scientific Research (No. 11640009)~Ministry of Education, Science, Sports and Culture of Japan. The second named author was supported in part by Grant-in-Aid for Scientific Research (NO.11640038), Ministry of Education, Science, Sports and Culture of Japan.

18


1.

INTRODUCTION

Throughout this paper except in the appendix, we denote by q a rational or an imaginary quadratic integer with Iq( > 1, and K an imaginary quadratic number field including q. Note that K must be of the form K = Q(q) if q is an imaginary quadratic integer. For a positive rational integer s and a polynomial P ( z ) E K [z] of degree s such that P ( 0 ) # 0, P(q-n) # 0 for all integers n _> 0, we define an entire function

Concerning the values of $(z; q), as a special case of a result of Bkivin [q, we know that if 4(a; q) E K for nonzero a E K , then a = a,q: with some integer n, where as is the leading coefficient of P ( z ) . Ht: usctl in the proof a rationality criterion for power series. Recently, the prcscnt authors [I] showed that n in B6zivin1s result must be positivct. Hcnc:c we know that, for nonzero CY E K ,

In case of P ( z ) = aszs + ao, it was also proved in [I] that &(a;q ) E K for nonzero a, E K if and only if a = asqsn with some n E N. In this paper we are interested in the particular case s = 2, P ( z ) = (z - q)2 of (1.1), that is,

On the values of certain q-hypergeometric series 11

where b is a nonzero rational integer and D is a square free positive integer satisfying

We now state our main result which completes the above result.

Theorem. Let q be a rational or an imaginary quadratic integer with 1q1 > 1, and K an zrnaginary quadratic number field including q. Let 4 ( ~q ); be the function (1.3). Then, for nonzero a E K , +(a; q) does not belong to K except when

where the order of each & sign is taken into account. Moreover, a, is a zero of 4(z; q) in each of these exceptional cases. For the proof, we recall in the next section a method developed in [I] and [2]. In particular, we introduce a linear recurrence c, = c,(q) ( n E N) having the property that +(qn; q) E K if and only if cn(q) = 0. Then the proof of the theorem will be carried out in the third section by determining the cases for which %(q) = 0. In the appendix we remark that one of our previous results (see Theorem 1 of [2]) can be made effective. The authors would like to thank the referee for valuable comments on refinements of an earlier version of the present paper.

2. We note that the function J ( z ; q) := +(-z2/4; q) satisfies

where the right-hand side is the Bessel function Jo(z). In this sense J ( z ; q) is a q-analogue of Jo(z). The main purpose of this paper is to determine the pairs (q, a ) with a E K for which 4(a;q) belong to K. In this direction we have the following result (see Theorems 2 and 3 of [2]): $(a,; q) does not belong to K for all nonzero a E K except possibly when q is equal to

19

A LINEAR RECURRENCE c,

Let +(z; q) be the function (1.1). Then, for nonzero a E K , we define a function

which is holomorphic at the origin and meromorphic on the whole complex plane. Since $(a; q) = f (q), we may study f (q) arithmetically instead of +(a; q). An advantage in treating f (z) is the fact that it satisfies the functional equation

which is simpler than the functional equation of d(z) = $(z; q) such as

20



where A is a q-'-difference operator acting as ( A $ ) ( z ) = $ ( q - l ~ ) .In fact, as a consequence of the result of Duverney [4], we know that f ( q ) does not belong to K when f ( z ) is not a polynomial (see also [ I ] ) . Since the functional equation (2.1) has the unique solution in K [ [ z ] ]a, polynomial solution of (2.1) must be in K [ z ] .Let E q ( P )be the set consisting of all elements a E K for which the functional equation (2.1) has a polynomial solution. Then we see that, for a E K ,

Note that f ( z ) r 1 is the unique solution of (2.1) with a = 0 , and that no constant functions satisfy (2.1) with nonzero a. In view of (1.2), o E E,(P)\{O) implies that a = a,qn with some positive integer n. ~ndeed,we can see that if (2.1) has a polynomial solution of degree n E N , then a must be of the form ar = a,qn. Hence, by (2.2), our main task is to determine the pairs ( q ,n ) for which the functional equation (2.1) with s = 2, P ( z ) = ( z - q ) 2 , and u = qn has a polynomial solution of degree n E N . To this end we quote a result from Section 2 of [2]with a brief explanation. ] (2.1) with s = 2, P ( z ) = Let f ( z ) be the unique solution in K [ [ z ]of ( z - q ) 2 , and a = qn ( n E N ) . It is easily seen that f ( z ) is a polynomial of degree n if and only if f ( z ) / P ( z )is a polynomial of degree n - 2. By setting n-2

21

Let Bn be an n x n matrix which is An with c as the last column. Since A, has the rank n - 1, this system of linear equations has a solution if and only if Bn has the same rank n - 1, so that det Bn = 0. We can show for det Bn ( n E N ) the recursion formula det Bn+2 = 29 det B n + ~ q2(1 - qn) det B,, with the convention det Bo = 0 , det B1 = 1. For simplicity let us introduce a sequence c, = c,(q) to be c, = q-("-')det B,, for which c1 = 1,c2 = 2, and

Then we can summarize the argument above as follows: The functional equation (2.1) with s = 2, P ( z ) = ( z - q ) 2 , and o = qn ( n E N ) has a polynomial solution f(z) if and only if c , ( q ) = 0 . We wish to show in the next section that c , ( q ) = 0 if and only if

which correspond to the cases given in the theorem.

3.

PROOF OF THE THEOREM

Let c, = ~ ( q ()n E N ) be the sequence defined in the previous section. The following is the key lemma for our purpose.

with unknown coefficients bi, we have a system of n - 1 linear equations of the form Anb = C ,

Lemma 1. Let d be a positive number. If the inequalities

where

and

(191 - ( 2

+ a)a-l)lqy r z(3 +s +a-l)

hold for some n = m, then (3.1) is valid for all n 2 m. Proof. We show the assertion by induction on n. Suppose that the desired inequalities hold for n with n 2 m. By the recursion formula (2.3) and the second inequality of (3.1), we obtain

which is the first inequality of (3.1) with n

+ 1 instead of n.

22



We next show the second inequality of (3.1) with n + 1 instead of n . By the recursion formula (2.3) and the first inequality of (3.1), we obtain

Noting that ( 2

+ 6 ) ( 2+ 6-'(Iqln + 1 ) ) is equal to

and that the inequality (3.2) for n = m implies the same inequality for all n 2 m, we get the desired inequality. This completes the proof. In view of the fact mentioned in the introduction, we may consider the sequences c, = c,(q) ( n E N ) only for q given just before the statement of the Theorem. In the next lemma we consider the sequence c,(q) for these q excluding b-.

Lemma 2. Let q be one of the numbers

We next consider the case where q = b

23

m without (1.4).

Lemma 3. Let b be a nonzero integer, and D a positive integer such that b2D 2 5. Then, for q = b m , the sequence c, = c,(q) ( n E N ) does not vanish for all n.

Proof. By (3.3), c3 and c4 are nonzero for the present q. Let us set A := b 2 ~ To . prove c, # 0 for all n 2 5, we show (3.1) and (3.2) with 6 = 3, n = 4. Indeed, by straightforward calculations, we obtain

and

Since these values are positive whenever A 2 5, (3.1) with 6 = 3, n = 4 holds. Moreover,

holds whenever A 2 5. Hence (3.2) with S = 3, n = 4 also holds. Hence the desired assertion follows from Lemma 1. This completes the proof.

0 Then, for the sequence c, = c,(q) ( n E N), c, = 0 if and only if (2.4) holds. Moreover, for the exceptional cases (2.4), 4(q3;q ) = 0 if q = -3, and 4(q4;q ) = 0 if q = (- 1 f f l ) / 2 , where +(z;q) is the function (1.3). Proof. Since c3 = 3 + q ,

C4

= 2(q2

+q +2),

we see that c, = 0 in the cases (2.4). By using computer, we have the following table which ensures the validity of (3.1) and (3.2) with these values:

It follows from Lemma 1 that, in each of the sequences, ~ ( q#)0 for all n 2 m. By using computer again, we can see the non-vanishing of the remaining terms except for the cases (2.4). As we noted in the previous section, if the functional equation (2.1) has a polynomial solution f ( z ;a ) , it is divisible by P ( z ) . Hence we have ~ # ( ~ ~=; fq( q) ;q 3 ) = 0 if q = -3, and 4 ( q 4 ; q ) = f (q;q4) = 0 if q = (-1 f -)/2. The lemma is proved. 0

By this lemma there remains the consideration of the case where q = bwith (1.4) and b2D < 5, that is the case q = fG.In this case, by using computer, we can show that (3.1) and (3.2) with 6 = 4, n = 8 are valid. Hence, by Lemma 1, c, = %(f # 0 for all n > 8. We

n)

see also that c, # 0 for all n < 8 by using computer again. Thus we have shown the desired assertion, and this completes the proof of the theorem.

Appendix Here we consider an arbitrary algebraic number field K , and we denote

by OK the ring of integers in K. Let d, h, and R be the degree over Q, the class number, and the regulator of K, respectively. Let s be a positive integer, q a nonzero element of K, and P ( z ) a polynomial in K [ z ]of the form S

Then, as in Section 2, we define a set &,(P) to be the set consisting of all a E K for which the functional equation (2.1) has a polynomial solution. In this appendix we remark that the following result concerning

24



the set Eq(P) holds. Hereafter, for any a E K, we denote by H ( a ) the ordinary height of a , that is, the maximum of the absolute values of the coefficients for the minimal polynomial of a over Z.

Theorem A. Let s be a positive integer with s 2 2, and q a nonzero element of K with q E OK or q-' E OK. Let ai(x) E OK[x], i = 0,1, ..., s , be such that

Let S = {wl, ...,wt) be the set of finite places of K for which lylwi < I , and B an upper bound of the prime numbers pl , ...,pt with \pi < 1. Let P ( z ) = P ( z ;q ) be a polynomial as above, where ai = ai(q) (i = 0, 1, ..., - 5 ) . Then there exists a positive constant C , which is effectively corr~putc~lle f7.f~V-Lquantities depending only on d , h, R, t , and B , such that i j E q ( r ) # { O ) , then H(q) C .

lwi

0 for 8 > 0 (use (r'/I?)'(s) = Ck.O(k + s ) - ~ ) ,we do have h(tm) < h(2) = (1 - n ( y + l o g r r ) ) / 2 + r l ( l -iog2) 5 (1 - n ( r + l o g r r - l+log2)/2 < 0.

30


The class number one problem for some non-normal sextic CM-fields

BOUNDS ON RESIDUES

2.3.

Lemma 3. 1. Let K be a sextic CM-field. Then, !j and CK (P) I 0 imply Ress=i(C~)2

1-P

< 1 - ( l j a log dK) 5 P < 1

where EK := 1 - ( 6 ~ e ' ~ ~ " l d g ~(3) ).

31

has a real zero p in [1 - (11log dN), l[. Then, First, assume that P 2 1 - (l/410gdK) (since [N : K ] = 4, we have dN 3 d k ) . Since CK(P) = 0 0, we obtain

and

(since [N : F] = 8, we have dN d i = fh6). Second assume that 1 such that FKIF = Qe. Since K / F is quadratic, for any totally positive algebraic integer a E F such that K = F ( 6 ) there exists some integral ideal Z of F such that (4a) = z23KIF(see [LYK]). In particular, there exists some integral ideal 1 of F such that (4ap) = ( 2 ) 2 ~ h := Z2FKIF = z2Qe and e is odd. 4. If (q) = F ( 6 ) . F(,/-) K would

3.1.

Q were inert in F , we would have K = F(-) = If (q) = Q3 were ramified in F , we would have K =

J-a3Q)

.

= ~(fi). In both cases, be abelian. A contradiction. Hence, q splits in F . = F(

= F(JT)

THE SIMPLEST NON-NORMAL SEXTIC CM-FIELDS

Throughout this section, we assume that h$ is odd. We let A F denote the ring of algebraic integers of F and for any non-zero a E F we let v(a) = f1 denote the sign of NFlq(a). Let us first set some no2 be any rational prime which splits completely in a tation. Let q real cyclic cubic field F of odd narrow class number h&, let Q be any one of the three prime ideals of F above q and let cup be any totally + positive generator of the principal (in the narrow sense) ideal Q h ~ We . set KFIQ:= F(J-CYQ) and notice that K F , is~ a non-normal CM-sextic field with maximal totally real subfield the cyclic cubic field F . Clearly, & is ramified in the quadratic extension K / F . However, this quadratic extension could also be ramified at primes ideals of F above the rational prime 2. We thus define a simplest non-normal sextic CM-field as being a K F I Qsuch that Q is the only prime ideal of F ramified in the quadratic extension K F I Q / F . Now, we would like to know when is K F ~ Q a simplest non-normal sextic field. According to class field theory, there is a bijective correspondence between the simplest non-normal sextic CM-fields of conductor Qe (e odd) and the primitive quadratic characters xo on the multiplicative groups (AF/Qe)* which satisfy xO(e)= v(c) for all e E U F (which amounts to asking that Z I+ ~ ( 1 =) v(az)xo(ar) be a primitive quadratic character on the unit ray class group of L for the + modulus Qe, where a= is any totally positive generator of z h r ) . Notice that x must be odd for it must satisfy x(-1) = v(-1) = ( - I ) ~ = -1. Since we have a canonical isomorphism from Z/qeZ onto AF/Qe, there

>

34



exists an odd primitive quadratic character on the multiplicative group ( A F / Q e ) * ,e odd, if and only if there exists an odd primitive quadratic character on the multiplicative group ( Z / q e Z ) * ,e odd, hence if and only if [q = 2 and e = 31 or [q = 3 (mod 4 ) and e = 11, in which cases there exists only one such odd primitive quadratic character modulo Qe which we denote by x p . The values of X Q are very easy to compute: for a E AL there exists a, E Z such that a = a, (mod Q e ) and we have x Q ( a )= xq(a,) where xu denote the odd quadratic character associated with the imaginary quadratic field Q(fi). In particular, we obtain:

is a simplest non-normal sextic CMProposition 7. K = F(,/-) field if and only if q $ 1 (mod 4) and the odd primitive quadratic character X Q satisfies x Q ( e ) = N F I Q ( € )for the three units € of any system of fundamental units of the unit group U F of F . I n that case the finite part of the conductor of the quadratic extension K F I Q / Fis given by

&>'K

e

1 8 ?r

.J

(log f F

35

,/Qf;

+ 0.05)2 log(Q12f i 6 )

(where BK := 1 - ( 6 ~ e ' / ~ ~ / d2$ e~ ~ ) := 1 - ( 6 ~ , 1 / 2 4 / 3 1 / 6 f l / ~ ) )In . particular, if h~ = 1 then f F 5 9 - lo5 and for a given fF 5 9 . lo5 we can use (10) to compute a bound on the Q's for which hk = 1. For example h~ = 1 and f~ = 7 imply 4 5 5 . 1 0 7 , hK = 1 and f F > 1700 imply 4 5 l o 5 , h~ = 1 and f~ > 7200 imply 4 5 lo4 and h K = 1 and fF > 30000 imply Q < lo3. Proof. Noticing that the right hand side of (10) increases with Q 2 3, we do obtain that f F > 9 lo5 implies h i > 1. Assume that h K F t q= 1. Then hg = 1, fF = 1 (mod 6 ) is prime or fF = 9, fF 9 - lo5, and we can compute BF such that (10) yields hKstq> 1 for q > B F (and we get rid of all the q 5 BF for which either 1 (mod 4 ) or q does not split in F (see Theorem 8 ) ) . Now, the q key point is to use powerful necessary conditions for the class number of KF,, to be equal to one, the ones given in [LO, Theorem 6) and in [ O h , Theorem 21. Using these powerful necessary conditions, we get rid of most of the previous pairs (q,f F ) and end up with a very short list of less than two hundred pairs (q,f F ) such that any simplest non-normal sextic number fields with class number one must be associated with one of these less than two hundred pairs. Moreover, by getting rid of the pairs ( q ,f F ) for which the modular characters X Q do not satisfy x Q ( e ) = N F / ~ ( tfor ) the three units e of any system of fundamental units of the unit group U F ,we end up with less than forty number fields K F l qfor which we have t o compute their (relative) class numbers. Now, for a given F of narrow , use the method developed in class number one and a given K F , ~we [Lou31 for computing h i F t q .To this end, we pick up one ideal Q above q and notice that we may assume that the primitive quadratic character x on the ray class group of conductor & associated with the quadratic extension K F , q / F is given by ( a ) I+ ~ ( c r )= v ( c r ) x Q ( a )where v ( a ) denotes the sign of the norm of cr and where X Q has been defined in subsection 3.1. According t o our computation, we obtained:

2 and @ = 23 if q = 2. T h e o r e m 9. Let K = K F l qbe any simplest nun-normal sextic CM-field. Then, QK = 1, dK = Qdg = ~ f : ,dN = Ql2d; = $d& = Q 12f F16, and ( 6 ) yields :

T h e o r e m 10. There are 19 non-isomorphic non-normal sextic CMfields K (whose maximal totally real subfields are cyclic cubic fields F ) which have class number one: the 19 simplest non-normal sextic CMin the following Table : fields K F , given ~



Table

I n this Table, fF is the conductor of F and F is also defined as being the splitting field of an unitary cubic polynomial (X) = x3- ax2+b~ - c with integral coeficients and constant term c = q which is the minimal polynomial of an algebraic element cuq E F of norm q such that KF,r= I?(,/%). Therefore, KFa is generated by one of the complex roots of ( X ) = -pF(-x2) = X6 a x 4 bX2 + C. the sextic polynomial PKF,,

+

+

References [Bou] G. Boutteaux. DBtermination des corps B multiplication complexe, sextiques, non galoisiens et principaux. PhD Thesis, in preparation. [CHI

P.E. Conner and J. Hurrelbrink. Class number parity. Series in Pure Mathematics. Vol. 8. Singapore etc.: World Scientific. xi, 234 p. (1988).

[LLO] F. Lemmermeyer, S. Louboutin and R. Okazaki. The class number one problem for some non-abelian normal CM-fields of degree 24. J. ThLor. Nombres Bordeaux 11 (lggg), 387-406.

37

S. Louboutin and R. Okazaki. Determination of all non-normal quartic CM-fields and of all non-abelian normal octic CM-fields with class number one. Acta Arith. 6 7 (1994)) 47-62. S. Louboutin. Majorations explicites de (L(1,x)I. C. R. Acad. Sci. Paris 316 (1993), 11-14. S. Louboutin. Lower bounds for relative class numbers of CMfields. Proc. Amer. Math. Soc. 120 (1994)) 425-434. S. Louboutin. Computation of relative class numbers of CMfields. Math. Comp. 66 (1997)) 173-184. S. Louboutin. Upper bounds on IL(1,x)J and applications. Canad. J. Math. 50 (1999)) 794-815. S. Louboutin. Explicit bounds for residues of Dedekind zeta functions, values of L-functions at s = 1 and relative class numbers. J. Number Theory, 85 (2000)) 263-282. S. Louboutin. Explicit upper bounds for residues of Dedekind zeta functions and values of L-functions at s = 1, and explicit lower bounds for relative class numbers of CM-fields. Preprint Univ. Caen, January 2000. S. Louboutin, Y.-S. Yang and S.-H. Kwon. The non-normal quartic CM-fields and the dihedral octic CM-fields with ideal class groups of exponent 5 2. Preprint (2000). R. Okazaki. Non-normal class number one problem and the least prime power-residue. In Number Theory and Applications (series: Develoments i n Mathematics Volume 2 ), edited by S. Kanemitsu and K. Gyory from Kluwer Academic Publishers (1999) pp. 273289. H.M. Stark. Some effective cases of the Brauer-Siege1 theorem. Invent. Math. 23 (1974)) 135-152. L.C. Washington. Introduction to Cyclotomic Fields. Grad. Texts Math. 83, second edition, Springer-Verlag (1997).

TERNARY PROBLEMS IN ADDITIVE PRIME NUMBER THEORY JBrg BRUDERN Mathematisches Institut A, Universitat Stuttgart, R- 70511 Stuttgart, Germany [email protected]

Koichi KAWADA Department of Mathematics, Faculty of Education, Iwate University, Morioka, 020-8550 Japan [email protected]

Keywords: primes, almost primes, sums of powers, sieves Abstract

+ +

We discuss the solubility of the ternary equations x2 y3 z k = n for an integer k with 3 5 k 5 5 and large integers n , where two of the variables are primes, and the remaining one is an almost prime. We are also concerned with related quaternary problems. As usual, an integer with a t most r prime factors is called a P,-number. We shall show, amongst other things, that for almost all odd n , the equation x2 +p: +p: = n has a solution with primes pl, p2 and a Pis-number x, and that for every sufficiently large even n , the equation x +p: pj: + p i = n has a solution with primes pi and a P2-number x.

+

1991 Mathematics Subject Classification: llP32, llP55, llN36, llP05.

1.

INTRODUCTION

The discovery of the circle method by Hardy and Littlewood in the 1920ies has greatly advanced our understanding of additive problems in number theory. Not only has the method developed into an indispensable tool in diophantine analysis and continues to be the only widely applicable machinery to show that a diophantine equation has many solutions, but also it has its value for heuristical arguments in this area.

'written while both authors attended a conference a t RIMS Kyoto in December 1999. We express our gratitude t o the organizer for this opportunity to collaborate. ..r

34

40

A N A L Y T I C NUMBER T H E O R Y

Ternary problems in additive prime number theory

This was already realized by its inventors in a paper of 1925 (Hardy and Littlewood [14]) which contains many conjectures still in a prominent chapter of the problem book. For example, one is lead to expect that the additive equation S

>

with fixed integers ki 2, is soluble in natural numbers xi for all sufficiently large n, provided only that

41

(1.3) and (1.4) with all variables restricted to prime numbers. With existing technology, we can, at best, hope to establish this for almost all n satisfying necessary congruence conditions. A result of this type is indeed available for the equations (1.3). Although the authors are not aware of any explicit reference except for the case k = 2 (see Schwarz [26]), a standard application of the circle method yields that for any k 2 2 and any fixed A > 0, all but O(N/(log N)*) natural numbers n N satisfying the relevant congruence conditions3 are of the form n = p: + p; + p;, where pi denotes a prime variable.

3 such that (p - 1)lk and p 3 (mod 4), this condition is equivalent to (i) n 1 or 3 (mod 6) when k is odd, (ii) n G 3 (mod 24), n f 0 (mod 5) and (n - l , q k ) = 1 when k is even but 4 { k, (iii) n E 3 (mod 24), n $ 0, 2 (mod 5) and (n - 1, qk) = 1 when 41k. It is easy to see that almost all n violating this congruence condition cannot be written in the proposed manner.

=

42



Theorem 3. Let N2 be the set of all odd natural numbers that are not congruent to 5 modulo 7. (i) For almost all n E N2, the equation x2 p; pi = n has solutions with a P3-number x and primes pl ,p2. (ii) For almost all n E N2,the equation p: y3 + p i = n has solutions with a P3-number y and primes pl, p2.

has solutions in primes pi and a P3-number x. (ii) For all suficiently Earge even n, the equation

+ + +

has solutions i n primes pi and a P4 -number x. Further we have a result when the linear term is allowed to be an almost prime.

It will be clear from the proofs below that in Theorems 1-3 we actually obtain a somewhat stronger conclusion concerning the size of the exceptional set; for any given A > 1 the number of n 5 N satisfying the congruence condition and are not representable in one of specific shapes, is lo^ lo^ N)-*). A closely related problem is the determination of the smallest s such that the equation S

43

Theorem 5. For each integer k with 3 5 k 5 5, and for all sufficiently large even n, the equation

has solutions i n primes pi and a P2-number x.

s

All results in this paper are based on a common principle. One first solves the diophantine equation at hand with the prospective almost prime variable an ordinary integer. Then the linear sieve is applied to the set of solutions. The sieve input is supplied by various applications of the circle method. This idea was first used by Heath-Brawn [15], and for problems of Waring's type, by the first author [3].

k=l has solutions for all large natural numbers n. This has attracted many writers since it was first treated by Roth [25] with s = 50. The current record s = 14 is due to Ford [8]. Early work on the problem was based on diminishing ranges techniques, and has immediate applications to solutions of (1.8) in primes. This is explicitly mentioned in Thanigasalam [27] where it is shown that when s = 23 there are prime solutions for all large odd n. An improvement of this result may well be within reach, and we intend to return to this topic elsewhere.

A simplicistic application of this circle of ideas suffices to prove Theorem 5. For the other theorems we proceed by adding in refined machinery from sieve theory such as the bilinear structure of the error term due to Iwaniec [19], and the switching principle of Iwaniec [18] and Chen [5]. The latter was already used in problems cognate to those in this paper by the second author [22]. Another novel feature occurs in the proof of Theorem 2 (ii) where the factoriability of the sieving weights is used to perform an efficient differencing in a cubic exponential sum. We refer the reader to $6 and Lemma 4.5 below for details; it is hoped that such ideas prove profitable elsewhere.

When one seeks for solutions in primes, one may also add a linear term in (1.8), and still faces a non-trivial problem. In this direction, Prachar [23] showed that is soluble in primes pi for all large odd n. Although we are unable to sharpen this result by removing a term from the equation, conclusions of this type are possible with some variables as almost primes. For example, it follows easily from the proof of Theorem 2 (ii) that for all large even n the equation

2.

has solutions in primes pi and a P4-number y. We may also obtain conclusions which are sharper than those stemming directly from the above results.

NOTATION AND PRELIMINARY RESULTS

We use the following notation throughout. We write e(a)= exp(2sicu), and denote the divisor function and Euler's totient function by r(q) and cp(q), respectively. The symbol x X is utilized as a shorthand for X < x < 5X, and N =: M is a shorthand for M

2 7 and h 2 2, or when p 5 5 and

Proof. The first two assertions are proved via standard arguments (refer to the proofs of Lemmata 2.10-2.12 of Vaughan [30] and Lemmata 8.1 and 8.6 of Hua [17]).The part (iii) is immediate from Lemma 3.1, since k j < 6 (0 5 j 5 s).

Lemma 3.3. Assume that

( p ,n). One also has T h e n one has B d ( p ,n ) = B(p,d)

Prwf. The assumption and Lemma 3.1 imply that Ad( p h ,n ) = A ( ~n)~= , 0 for h > k , thus

(3.1)

The series defining B d ( p ,n) and B ( p , n ) are finite sums in practice, because of the following lemma.

Lemma 3.1. Let B(p, k ) be the number such that power of p dividing k , and let

(iii) A d ( p ,n ) = A ( p , n ) = 0 , when p h 5.

49

d ) k ) , which For h 5 k , we may observe that s k ( p h ,a d k ) = s k ( p h , gives A ~ (n ) ~= A ~ ( ,~ ,( p~h), n). SO the former assertion of the lemma follows from (3.4). Next we have

is the highest

when p = 2 and k is even, when p > 2 or k is odd.

(3.3)

+

T h e n one has S; ( p h ,a ) = 0 when p a and h > y ( p , k ) .

But, when 1 5 h 5 k , we see

Proof. See Lemma 8.3 of Hua [17] Next we assort basic properties of A d ( q ,n ) et al.

Lemma 3.2. Under the above convention, one has the following. (i) A d ( q ,n) and A ( q , n) are multplicative functions with respect to q.

(ii) B d ( p ,n ) and B ( p , n ) are always non-negative rational numbers.

whence

1 -Ap(ph,n) = (1 - ; ) ~ ( p " , n ) . Al(ph,n)- P

Obviously the last formula holds for h = 0 as well. Hence the latter assertion of the lemma follows from (3.4).

50


Ternary problems i n additive prime number theory

51

Now we commence our treatment of the singular series appearing in our ternary problems, where we set s = 1.

Lemma 3.4. Let Ad(q, n) be defined by (2.2) with s = 1, and with natural numbers k, ko and kl less than 6. Then for any prime p with p f d, one has /Ad(P, n) 1

< 4kkOklp-l (P, n ) 'I2-

When pJd one has

By appealing to (3.5) and (3.6), a straightforward estimation yields

Proof. For a natural number 1, let Al be the set of all the non-principal Dirichlet characters x modulo p such that XL is principal. Note that

For a character

x modulo p and an integer m, we write

As for the Gauss sum T(X,I ) , we know that IT(x, 1)1 = p1I2, when x is non-principal. It is also easy to observe that when x is non-principal, we have T(X,m) = ~ ( m ) r ( x1). , When x is principal, on the other hand, we see that T(X,m ) = p - 1 or -1 depending on whether plm or not. In particular, we have

for any character x modulo p and any integer m. By Lemma 4.3 of Vaughan [30], we know that

whenever p { a, and obviously Sf (p, a ) = Sl (p, a ) - 1. So when p { d, we have Sk(p, adk) = Sk(p, a ) and

When pld, we have Sk(p,adk) = p, and the proof proceeds similarly. By using (3.5), (3.6) and (3.7), we have

Thus when pld but p t n, we have Ad(p,n ) = o ( ~ - ' / ~ by ) (3.6). When pln, we know T ( + ~ +-n) ~ , = 0 unless $o+l - is principal, in which case we have T ( + ~ +-n) ~ , = p - 1 and $1 = $o, and then notice that +o E A ( k o , k l ) , because both of and are principal. Therefore when pld and pln, we have

$2

$tl

by (3.5), and the proof of the lemma is complete.

Lemma 3.5. Let Ad(q,n), Gd(n,L ) and Bd(p,n) be defined by (2.2)) (2.3) and (3.2)) respectively, with s = 1, and with natural numbers k, ko and kl less than 6. Moreover put Y = exp( d m ) and write

52



Proof. We define Q to be the set of all natural numbers q such that every prime divisor of q does not exceed Y, so that we may write

in view of Lemma 3.2 (i). We begin by considering the contribution of integers q greater than N1I5 to the latter sum. Put q = 10(log N ) - ' / ~ . Then, for q > N1I5, we see 1 < ( q / ~ ' / ~ )=' q V y - 2 ,and

for all primes p with p whence Td(p,a )

l/(qlq2) Unless ql = 92 and a1 = a?, we have Ila2lq2-allqlll where IlPll = minmEzIP - ml, so the last expression is

>N - ~ / ~ ,

which means that there is an absolute constant C > 0 such that

Therefore a simple calculation reveals that by using (3.13). Consequently we obtain the estimate

which yields

Next we consider the sum The lemma follows from (3.9), (3.10) and the last estimate. adk)si0(q,a)Sil (q, a) for short. By Now write Td(q,a) = q-1rp(q)-2~k(q, (3.7) and (3.6) we have S k (p, adk) q/N we take coprime integers r and b such that Jrcr- bl J q a- a ) / 2 and r 5 2/)qa - al, according to Dirichlet's theorem again. Since b/r cannot be identical with a/q now, we see

> 6 (n)J ( n ) (log N)- . This completes the proof of Theorem 5.

'

We next turn to Theorem 1 and Theorem 2 (i). At this stage we benefit from the bilinear error term in Iwaniec's linear sieve. Without it, we can prove Theorems 1 and 2 (i) only with P16and P7,respectively, at present. Thus we require Iwaniec's bilinear error terms within a weighted sieve. This has been made available by Halberstam and Richert [13]. Rather than stating here their result in its general form, we just mention its effect on our particular problem within the proof below. Indeed the inequality (5.27) below is derived from Theorems A and B of Halberstam and Richert [13] (see also the comment on (8.4) in [13], following Theorem B), by taking U = T = 213, V = 114 and E = 119, for example. Here we should make a minor change in the error term in Theorem A of [13], where the bilinear error term is expressed by using the supremum over all the sequences (X,) and (p,) satisfying IX,I 5 1 and lpvl 5 1, instead of the form appearing in (5.27). This change is negligible in the argument of Halberstam and Richert [13], and as the following proof shows, it is convenient for our aim to keep the error term in the original form given by Iwaniec [19]. In order to establish (5.27) below, we may alternatively combine Richert's weighted sieve with Iwaniec's linear sieve, while Halberstam and Richert [13] appealed to Greaves' weighted sieve. Although Greaves' weights give stronger conclusions, Richert's weights are simpler, and much easier to combine with Iwaniec's sieve. It is actually a straightforward task to utilize Iwaniec's sieve within the proof of Theorem 9.3 of Halberstam and Richert [12], and such a topic is discussed in $6.2 of the unpublished lecture note [20] of Iwaniec. The latter device is still adequate to our purpose, proving (5.27). We may leave the details of the verification of (5.27) to the reader.

72


Ternary problems i n additive prime number theory

Before launching the proofs of Theorems 1 and 2 (i), we record a simple fact as a lemma. Lemma 5.1. Let v(n) be the number of distinct prime divisors of n, and A be any constant exceeding 3. Then one has v(n) 2Aloglog N for all but O(N(log N ) - ~ )natural numbers n 5 N .

for all primes p and integers 1 1. In fact, we can show the former by Lemmata 3.3 and 3.6 (i) and (ii), following the verification of (5.10). As regards the latter, it suffices to note that, in the current situation, we always have

by Lemmata 3.1 and 3.4, respectively. We next discuss a lower bound for Pl(n, Y). By using (5.18)) and by combining the former formula in (5.21) with Lemma 3.4, we have

PI(n, Y)

>

n

(zp2)-'

n

(1 - 120p-')

n

(I - 1 2 0 ~ - ' / ~ ) ,

74

ANALYTICNUMBER THEORY


75

for n E M(kl ) . If we denote by fin the v(n)-th prime exceeding lo5 where v(n) is defined in Lemma 5.1, then we have

We shall estimate E(n) on average over n. To this end, we first estimate the expression

as well as 0, and that

2e7 2e7 f o r 0 < u < 3 . ) go(,) = -log(u - 1) for 2 5 u 5 4, + 1 ( ~ = U U (6.1)

76


ANALYTIC NUMBER T H E O R Y

Then we refer to Iwaniec's linear sieve in the following form (see Iwaniec [19], or [20], for a proof). Lemma 6.1. Let Q, U, V, X be real numbers >_ 1, and suppose that D = UV is suficiently large. Let w(d) be a multiplicative function such that 0 w(p) < p and w ( ~ ' )< 1 for all primes p and integers 1 1, and suppose that -l

< logz - log w

wsp

(6.2)

whenever z > w 2 2. Further, let r(x) be a non-negative arithmetical function, z be a real number with 2 z II(z) be as in (2.11)) and write

<

Y, thus

e = P ( n , q7 (1 log z

+ O((l0g2')-1)).

Hence it follows from (6.27), (6.26) and (6.1) that

85

of n E N2n[N, (615)N]" , where N2 is the set introduced in the statement of Theorem 3. When X and y are sets of integers, we denote by R(n, X , y ) the number of representations of n in the form n = x2 y3 p3 subject to x E X , y E Y and primes p We denote by X(d) and Y(d), respectively, the set of the multiples of d in the intervals (X2,5X2) and (X3, 5X3), and by Xo and Yo, respectively, the sets of primes in the intervals (X2,5x2) and (X3, 5x3). We further put

-

+ +

and define the sets

for almost all n. Here we can confirm the numerical estimates

and C7(7) = 0, as is recorded in (12) of Kawada [22]. (To confirm these bounds, we can appeal to Theorem 1 of Grupp and Richert [9]. We remark that the function IT (u) in [9] and our CT(u) satisfy the relation C T ( 4 = UIT(U).) Finally, by (6.29) and (6.30), we obtain the estimate

Trivially, all the numbers in X3 and y3 are P3-numbers. By orthogonality, we have the formulae

1

R(n) X2, y(d)) = for almost all n. We conclude that R(n) - ~ ( n >) 0 for almost all n , by (6.17) and the last inequality, and the proof of Theorem 2 (ii) is now completed. We proceed to the proofs of Theorems 3 and 4, following the methods introduced thus far. These proofs are somewhat simpler than that of Theorem 2 (ii) above. Concerning Theorem 3, moreover, the limits of "level of distribution" D assured by Lemmata 4.3 and 4.4 are not worse than those appearing in Lemmata 4.5 and 4.2, and the latter constraints were still adequate for getting a "P3"as we saw in the above proof of Theorem 2 (ii), whence the conclusions in Theorem 3 are essentially obvious to the experienced reader. For these reasons, we shall be brief in the following proofs. The proof of Theorem 3. Within this proof, we use the terminology "for almost all n", for short, to mean that "for all but O(N(1og N ) - ~ )values

(x 7

T =4

g2(a; ~ 2 . 7 , ~f3(a; ) ) d)g3 ( a ; xil')e(-no)do.

to these integrals are immediThe contributions from the major arcs ately estimated by Lemma 2.1, with the aid of Lemma 2.2 in the latter case. As in the proof of Theorem 2 (ii), then we can apply Iwaniec's linear sieve, Lemma 6.1, to obtain a lower estimate for R(n, XI, yo) and an upper estimate for R(n, X2,y l ) , both valid for almost all n. Now let I ( n ) and P ( n , Y) be defined by (2.4), (3.1), (3.2) and (6.15), with s = 1, k = 2 and ko = kl = 3, and put

By Lemmata 3.1, 3.3, 3.4, 3.6 and 5.1, we may show that

for almost all n , and Lemma 2.1 asserts that I ( n ) = N1lg(log N ) - ~for N 5 n 5 (6/5)N.

-

T'

86



When we apply Lemma 6.1 to R(n, XI, yo), we are concerned with a remainder term corresponding to in Lemma 6.1, with U = D2I3 and V = 0'i3. We can regard this remainder term as negligible for almost all n by Lemmata 3.5 and 4.3. Using Lemma 3.3 in addition, we can establish the lower bound

The proof of Theorem 4. Let k2 = 4 or 5, and put

and r(4)=3,

log D

> (40(-)log z

+

87

r(5)=4.

We denote by R(n) the number of representations of n in the form

log log N)-'I5'))

~ ( nY),

log z (log X2)I ( n ) subject to primes pl, p2, p3 and integers x satisfying

for almost all n. On the other hand, we apply Lemma 6.1 to R(n, X2, y l ) , with taking U = Dl and V = 312, and dispose of the error term corresponding to d l ) in Lemma 6.1 by Lemmata 3.5 and 4.4. In this way we arrive at the upper bound log Dl R(n, X2, Yl) < (41 log z'

(-)

+ loglog log N)-'I5'))

e-7

P ( n , Y)logz'

x

Also we denote by ~ ( n the ) number of representations counted by R(n) with the additional constraint R(x) > r(k2). We aim to prove Theorem 4 by showing that R(n) - ~ ( n >) 0 for every even integer n E [N, (615)N]. We first fix some notation. Let I(n) and B(p, n) be defined by (2.4), (3.1) and (3.2) with s = 2, k = 1, ko = 2, kl = 3 and k2 = 4 or 5, and

+

which is valid for almost all n. Since yo c Yl, we see R(n, X2, Yo) 5 R(n, X2, yl), and

for almost all n , by (6.31), (6.32) and (6.30) with modest numerical computation. This proves Theorem 3 (i). Similarly we can show that

R(% XI 7 Y2)

2, so we see by (6.35) that

whence 2K (4)/8' < 5. Since (2 log 2)/8(4) > 5.5, we conclude by (6.33) and (6.34) that R(n) - ~ ( n 2) R(n) - R1(n) > 0 for every even n E [N, (6/5) N], which establishes Theorem 4 (i) . In the case k2 = 5, we have C3(3/0(5)) > 4.9 and C4(3/0(5)) > 4, thus we see by (6.35)

whence 2K(5)/01 < 13.5. Since (2 log 2)/0(5) > 14, we conclude by (6.33) and (6.34) that R(n) - ~ ( n > ) 0 again for every even n E [N, (615)N], which completes the proof of Theorem 4 (ii).

References

To examine K(k2), the following observation is useful. For a fixed real number u exceeding 1 and a large real number X , let x(X, u) be the number of integers x with x X and (5, II(xl/.)) = 1. Then we have

-

by Lemma 2.2 (with the latter formula in (2.9) for r = 1). On the other hand, to estimate x(X,u) is the simplest linear sieve problem, and it is easy to prove that

[I] J . Briidern, Iterationsmethoden in der addi tiven Zahlentheorie. Thesis, Gijttingen 1988. [2] J . Briidern, A problem in additive number theory. Math. Proc. Cambridge Philos. Soc. 103 (1988), 27-33. [3] J. Briidern, A sieve approach to the Waring-Goldbach problem I: Sums of four cubes. Ann. Scient. EC. Norm. Sup. (4) 28 (1995), 46 1-476. [4] J. Briidern and N. Watt, On Waring'sproblem for four cubes. Duke Math. J . 77 (1995), 583-606. [5] J.-R. Chen, On the representation of a large even integer as the sum of a prime and the product of a t most two primes. Sci. Sinica 16 (1973), 157-176. [6] H. Davenport and H. Heilbronn, On Waring's problem: two cubes and one square. Proc. London Math. Soc. (2) 43 (1937), 73-104.

r v

90



[7] H. Davenport and H. Heilbronn, Note on a result in additive theory of numbers. Proc. London Math. Soc. (2) 43 (1937), 142-151. [8] K. B. Ford, The representation of numbers as sums of unlike powers, 11. J. Amer. Math. Soc. 9 (1996), 919-940. [9] F. Grupp and H.-E. Richert, The functions of the linear sieve. J. Number Th. 22 (1986), 208-239. [lo] H. Halberstam, Representations of integers as sums of a square, a positive cube, and a fourth power of a prime. J. London Math. Soc. 25 (1950), 158-168. [ll]H. Halberstam, Representations of integers as sums of a square of a prime, a cube of a prime, and a cube. Proc. London Math. Soc. (2) 52 (1951), 455-466.

[12] H. Halberstam and H.-E. Richert, Sieve Methods. Academic Press, London, 1974. [13] H. Halberstam and H.-E. Richert, A weighted sieve of Greaves' type 11. Banach Center Publ. Vol. 17 (1985), 183-215. [14] G. H. Hardy and J. E. Littlewood, Some problems of "Partitio Numerorum": VI Further researches in Waring's problem. Math. Z. 23 (1925), 1-37. [15] D. R. Heath-Brown, Three primes and an almost prime in arithmetic progression. J. London Math. Soc. (2) 23 (l98l), 396-414. [I61 C. Hooley, On a new approach to various problems of Waring 's type. Recent progress in analytic number theory (Durham, 1979), Vol. 1. Academic Press, London-New York, 1981, 127-191. [17] L. K. Hua, Additive Theory of Prime Numbers. Amer. Math. Soc., Providence, Rhode Island, 1965. [18] H. Iwaniec, Primes of the type 4(x, y) form. Acta Arith. 21 (1972), 203-224.

A

+ A, where 4 is a quadratic

[19] H. Iwaniec, A new form of the error term in the linear sieve. Acta Arith. 37 (1980), 307-320. [20] H. Iwaniec, Sieve Methods. Unpublished lecture note, 1996. [21] W. Jagy and I. Kaplansky, Sums of squares, cubes and higher powers. Experiment. Math. 4 (l995), 169-173. [22] K. Kawada, Note on the sum of cubes of primes and an almost prime. Arch. Math. 69 (1997), 13-19. [23] K. Prachar , ~ b e ein r Problem vom Waring-Goldbach 'schen Typ 11. Monatsh. Math. 57 (1953) 113-116.

91

[24] K. F. Roth, Proof that almost all positive integers are sums of a square, a cube and a fourth power. J . London Math. Soc. 24 (1949), 4-13. [25] K. F. Roth, A problem in additive number theory. Proc. London Math. Soc. (2) 53 (1951), 381-395. [26] W. Schwarz, Zur Darstellung von Zahlen durch Summen von Primzahlpotenzen, 11. J. Reine Angew. Math. 206 (l96l), 78-1 12. [27] K. Thanigasalam, On sums of powers and a related problem. Acta Arith. 36 (1980), 125-141. [28] R. C. Vaughan, A ternary additive problem. Proc. London Math. SOC.(3) 41 (1980), 516-532. [29] R. C. Vaughan, Sums of three cubes. Bull. London Math. Soc. 17 (1985), 17-20. (301 R. C. Vaughan, The Hardy-Littlewood Method, 2nd ed., Cambridge Univ. Press, 1997.

A GENERALIZATION OF E. LEHMER'S CONGRUENCE AND ITS APPLICATIONS Tianxin CAI Department of Mathematics, Zhejiang University, Hangzhou, 31 0028, P.R. China

Keywords: quotients of Euler, Bernoulli polynomials, binomial coefficients Abstract

In this paper, we announce the result that for any odd n (n-1)/2 1 -292 (n)

+ nq:(n)

> 1,

(mod n2),

where g,(n) = (r4(n)- l ) / n , (r,n) = 1 is Euler's quotient of n with base r , which is a generalization of E. Lehmer's congruence. As applications, we mention some generalizations of Morley's congruence and Jacobstahl's Theorem to modulo arbitary positive integers. The details of the proof will partly appear in Acta Arithmetica.

2000 Mathematics Subject Classification: 1lA25, 1lB65, 1lB68.

1.

INTRODUCTION

In 1938 E. Lehmer [5] established the following congruence:

for any odd prime p, which is an improvement of Eisenstein's famous congruence (1850):

Partly supported by the project NNSFC..

93 C. Jia and K. Matsumoto (eak)!Analytic Number Theory, 93-98.

94

A generalization of E. Lehmer's congruence and its applications

ANALYTICNUMBER THEORY

where

95

Corollary 2. If n is odd, then (n-

is Fermat's quotient, using (1) and other similar congruences, he obtained various criteria for the first case of Fermat's Last Theorem (Cf. [8]). The proof of (1) followed the method of Glaisher 121, which depends on Bernoulli polynomials of fractional arguments. In this paper, we follow the same way t o generalize (1) to modulo arbitary positive integers, however, we need establish special congruences concerning the quotients of Euler. The main theorem we obtain is the following,

Theorem 1. If n

1112 2

f2q2(n)vn + q;(n)

(mod n),

where the negative sign is to be chosen only when n is not a prime power. In 1895, Morley [7] showed that

> 1 is odd, then

(n-1)/2

7

-292(n)

+ ng$(n)

>

for any prime p 5, this is one of the most beautiful congruences concerning binomial coefficients. However, his ingenious proof, which is based on an explicit of De Moivre's Theorem, cannot be modified to investigate other binomial coefficients, we use Theorem 1 to present a generalization of (2), i.e.,

(mod n2),

Theorem 3. If n is odd, then is Euler's quotient of n with base r .

4nl4 (-l)qn((&;/2)

Corollary 1. If n is odd, then

=4 m

-

dln

1

7 = 9 2 ( 4 - nq:(n)/2 i= 1

(mod n2)

for

3t n

(mod n2/3)

for

3 1 n.

Similarly as Theorem 1,we can generalize other congruences by Lehmer t o modulo arbitary positive integers. Among those, the most interesting one might be the following,

Theorem 2. If n is odd, then

(mod n3) for (mod n3/3) for

3 1n 3 ( n,

(3) where p(n) is Mobius' function, and +(n) is Euler's function. I n particular, if n 2 5 is prime, (3) becomes (2).

Corollary 3. If p

2 5 is prime, then

and

for any1 2 1. where

Corollary 4. For each 1 >_ 1, there are exactly two primes up to 4 x 1012 such that

is generalized Wilson's quotient or Gaussian quotient, the negative signs are to be chosen only when n is not a prime power.

the positive sign is to be chosen when p = 1093 and the negative sign is to be chosen when p = 3511.

96


A generalization of E. Lehrner's congruence and its applications

Corollary 5. If p, q 2 5 are distinct odd primes, then

for any prime p 2 5. This is a consequence of Jacobstahl's Theorem, and therefore a consequence of Theorem 4. The exponent 3 in (5) can be increased only if plBp-3, here Bp-3 is the p - 3th Bernoulli number. Jones (Cf. [3]) has asked for years that whether the converse for (5) is true. As direct consequences of Corollary 7 and Corollary 8, we present two equivalences for Jones' problem, i .e.,

Moreover, we have the following,

Theorem 4. Let n

> 1 be an integer, then ( (mod n3)

dln

for any integers u becomes

(

97

Theorem 5. If the congruence if 3 1. n , n # 2a

(modn3/3) 2 i 3 i n (mod n3/2) if n = 2a, a (mod n3/4) i f n = 2

> v > 0. I n particular, if p

>2

> 5 is prime,

(

)

1 (mod n3)

(4)

has a solution of prime power pi ( 1

( )

then ( 4 )

> l ) , then p must satisfy 1 (mod p6).

The converse is also true. Meanwhile, if the congruence (6) has a solution of product of distinct odd primes p and q, then this is Jacobstahl's Theorem. Corollary 6. If p, q

> 5 are distinct primes, then

( )

1 (mod p3),

(

)

1 (mod g3)

The converse is also true. In particular, if 1 = 2, the first part of Theorem 5 was obtained by R. J. McIntosh [6] in 1995.

for any integers u > v > 0. Corollary 7. If p

> 5 is prime,

then

Acknowledgments The author is very grateful to Prof. Andrew Granville for his constructive comments and valuable suggestions.

and

References

for any 12 1. Corollary 8. If p, q

> 5 are distinct primes, then

In 1862, Wolstenholme showed that

(

)

1 (mod p3)

[I] T. Cai and A. Granville, O n the residue of binomial coeficients and their products modulo prime powers, preprint . [2] J. W. L. Glaisher, Quart. J. Math., 32 (1901), 271-305. [3] R. Guy, Unsolved problems i n number theory, Springer-Verlag, Second Edition, 1994. [4]G. H. Hardy and E. M. Wright, A n introduction to the theory of numbers, Oxford, Fourth Edition, 1971. [5] E. Lehmer, O n congruences involving Bernoulli numbers and the quotients of Fermat and Wilson, Ann. of Math., 39 (1938), 350359.

98


[6] R. J. McIntosh, On the converse of Wolstenholmes theorem, Acta Arith., 71 (1995), 381-389. [7] I?. Morley, Note on the congruence 2*" = (-1)"(2n)!l(n!)~, where 2n 1 is prime, Ann. of Math., 9 (1895), 168-170. [8] P. Ribenboim, The new book of prime number records, SpringerVerlag, Third Edition, 1996.

+

ON CHEN'S THEOREM CAI Yingchun and LU Minggao Department of Mathematics, Shanghai University, Shanghai 200436, P. R. China

Keywords: sieve, application of sieve method, Goldbach problem Abstract

Let N be a sufficiently large even integer and S ( N ) denote the number of solutions of the equation

where p denotes a prime and Pz denotes an almost-prime with a t most two prime factors. In this paper we obtain

where

2000 Mathematics Subject Classification: 1lNO5, 1lN36.

1.

INTRODUCTION

In 1966 Chen ~ i n ~ r u n [ made 'I a considerable progress in the research of the binary Goldbach conjecture, heI21 proved the remarkable Chen's Theorem: let N be a sufficiently large even integer and S ( N ) denote the number of solutions of the equation

where p is a prime and P2 is an almost-prime with at most two prime factors, then

Project supported by The National Natural Science Fundation of China (grant no.19531010, 19801021) 99

C. Jia and K. Matsumoto (eds.), Analytic Number Theory, 99-1 19.

100


where

On Chen's theorem

101

where w(d) is a multiplicative function, 0 5 w(p) < p, X is independent of d. Then

The oringinal proof of Chen's was simplified by Pan Chengdong, Ding Xiaqi, Wang ~ u a n w Halberst , a m - ~ i c h e r t [ ~~]a, l b e r s t a m [ ~~]o, s s [ ~In l. [4] Halberstam and Richert announced that they obtained the constant 0.689 and a detail proof was given in (51. In page 338 of [4] it says: "It would be interesting to know whether the more elaborate weighting procedure could be adapted to the numerical improvements and could be important". In 1978 Chen ~ i n ~ r u nintroduced [~l a new sieve procedure to show

where log D log z '

s=-

RD =

V(z) = c ( w ) E (1 log z In this paper we shall prove

C(W) =

Irdlr

d O),

C

eE E

1

ep3 =N ( d ) 1

dl,

where Let

then

R1=

C d lD

N $ 0 and for all T > To,

Thus it is clear that the following is equivalent to Montgomery's pair correlation conjecture.

Conjecture. For all T > To and for any a, > 0, we have

for T # y,

where the argument is obtained by the continuous variation along the iT, starting with the value zero. straight lines joining 2, 2 + iT, and When T = y, we put

4+

The above argument has been noticed in pp. 242-243 of h j i i [5] with slightly more details. Thus it may be realized that to study the sum Then the well known Riemann-von Mangoldt formula (cf. p.212 of Titchmarsh [12]) states that

where 8 ( T ) is the continuous function defined by

is very important. Under the assumption of the Riemann Hypothesis, we have shown in Fujii [4] that for all T > To and for any positive a 0, we have

>

T

T log - . S(x, $) 27r 27r

=-

where T > To and a is any positive number. To state our conjecture and the results on this, we have to start by stating the following Riemannvon Mangoldt formula for N(T, x), the number of the zeros of L ( s , X) in 0 5 X ( s ) 5 1 and 0 5 S ( s ) = t 5 T, possible zeros with t = 0 or t = T counting one half only. Then it is well-known (cf. p.283 of Selberg [Ill) that for all T > 0, we have

where S(T, X) is defined as in p.283 of Selberg [ll]. Now if x = $, then the asymptotic behavior of the above sum must be similar to the one conjectured in Montgomery's conjecture. On the other hand, if x # $, then it may be natural to suppose that there may be no strong tendency to separate "(x) and To and for any a > 0, we have

C

00

Example 2. Let q be a non-negative integer, let f ( x ) = aox9+' +alx9+ . a g x Px9 log x , where ai E R (i = 0, . . . ,q ) , 0 < P E R and a0 is the irrational number of finite type q. Then f ( x ) satisfies the conditions of Theorem 3:

+

holds whenever a is the irrational number of finite type q and p # 0. The following theorem includes as a special case the above result (1.2).

145

+

f ('+')(x)

= (q

+ I ) ! a0 + pcx-'

for some c

> 0,

Theorem 1. Let f ( x ) be a twice differentiable function defined for x 2 1. Suppose that there exists an irrational number a of finite type q such that for x 1 1 either

and f ' ( x ) = a + 0(1 f " ( ~ ) ] ' / ~Then ) . for any D N ( f ( n ) )0

Theorem 4. Let a be an irrational number of constant type and let q, Q, and f ( x ) be as i n Theorem 3. Then I

DN ( f ( n ) )< N - 2 ~ 2 - 1 log / 2 N.

146

2.


Discrepancy of some special sequences

SOME LEMMAS

B y Abel's summation formula, we have

First, Erdos-TurBn's inequality is stated (see [2,p. 1141).

Lemma 1 (Erdos-Tursn's inequality). For any finite sequence of real numbers and any positive integer m , it follows

(I,)

Subsequently, two known results are stated (see [5, p.74, Lemma 4.71 for the first one and [6,p.226, Lemma 10.51 for the second one).

Lemma 2 (van der Corput). Let a and b be real numbers with a < b. Let f ( x ) be a real-valud function with a continuous and steadily decreasing f l ( x ) in ( a ,b ) , and let f l ( b ) = a , fl(a) = 8. Then

I t follows that

From (2.2),it follows that in each of the intervals where q is any positive constant less than 1.

Lemma 3 (Salem). Let a and b be real numbers with a < b. Let r ( x ) be a positive decreasing and differentiable function. Suppose that f ( x ) is a real-valued function such that f ( x ) E c 2 [ a b, ] , f " ( x ) is of constant sign and r l ( x ) /f " ( x ) is monotone for a 5 x 5 b. Then

there exists at most one number of the form 11 jail, 1 5 j 5 h, with no such number lying in the first interval. Therefore, we have

We need the following inequality.

Lemma 4. Let q be a non-negative integer, and let Q = 29. If irrational number of type < d , then for any positive integer m

0, a is of type < d with $(q) = cq"1+6/2 for some c > 0. Then, Lemma 4 with q = 0 implies

PROOF OF THEOREMS

Proof of Theorem 1. Let h be a positive integer. Applying Lemma 2, we get

m

Applying Lemma 1, by (3.1) and (3.2), we obtain where A = h f ' ( N ) and B = h f l ( l ) . We set g ( x ) = h( f ( x ) - a x ) . Using integration by parts, we have

for any 6 > 0. Choosing m =

Hence,

In the case f l ( x ) < a , f " ( x ) same lines as above.

> 0 for x 2 1, the proof runs along the 0

Proof of Theorem 2. The proof runs the same lines as in the proof of Theorem 1. Since a is of constant type, by Lemma 4 with q = 0 and + ( x ) = C , we have We suppose that f l ( x ) > a and f " ( x )

< 0 for x

> 1.

From Lemma 3 and the hypothesis, it follows that

1 h= 1

h1/211hall

a for x 2 1, then J : . - - J : f(q+')(x h . t ) d t l ...dt, > a ( x 1). It follows from Lemma 3 that

for any positive integer m. Choosing m = [N2I3], we have

>

+

which completes the proof. J1

Proof of Theorem 3. The case q = 0 coincides with Theorem 1, and so, we may suppose q 2 1. Let 1 5 H 5 N. Set Hq = H, Hi-1 = H : ~ ~ (2 = q, q - 1, . . . , 2). Applying Lemma 5, for any positive integer k we have

I$ where

-n ; hence for c @ -No by analytic continuation), it follows from (3.1), (4.1)) (4.3), (4.4) and (4.6) that it is easily seen that

170


Pad6 approximation t o the logarithmic derivative

...

171

where Similarly the leading coefficient of Qn-l(z) is P u t f (z) = F(a, b, c ; z) and g(z) = F(-a, -b, 1 - c ; z) for brevity. Then it follows that

from which we get $I

= f'g

1 - 22 + fg' - ab f19'-

z ( l - z) ab (f "gt

+ f 2").

We now use the differential equation satisfied by the Gauss hypergeometric function, stated in Section 1, that is,

Finally we propose the following interesting problems:

Problem 1. Find a continued fraction expansion of H(a, b, c ; z) analogous to Gauss' continued fraction expansion to G(a, b, c ; 2). Problem 2. Extend our theorem to the generalized hypergeometric function

and

in order to show that $(z) = 0 ; hence $(z) = 1 by $(O) = 1. Therefore &(z) is equal to F ( a n 1,b n 1,c 2n 1; z), as required. Finally the assumption that a , b $! -No and c $! N can be easily removed by limit operation, since each Aijn,BjYfand Cn are continuous at a , b E -No and at c E N. This completes the proof of the theorem.

+ +

5.

+ +

+ +

where p 5 q

+ 1 and ai E @, bj $! -No.

The simple combinatorial method employed in this paper will be certainly applied to construct Pad6 approximation of the second kind for the generalized hypergeometric function and its derivatives. Such approximations may have an interesting application to the irrationality problem of the ratio

REMARKS AND OPEN PROBREMS

It is easily seen from (1.1) that H ( a , b, c ;z) is a rational function if and only if Cn = 0 for some n E N. Therefore the explicit expression of Cn implies that H(a, b, c ; z) is a rational function if and only if either a E - N , ~ E- N , c - a E -No or c - b ~-No. Using the same method employed in Section 3, we can calculate the leading coefficient of the polynomial Pn(z) ; namely,

for q E Z \{O}, where n # rn E N and Ln(z) = polylogarithm of order n .

xp?lzk/kn is the r

References [I] G. V. Chudnovsky, Pad6 approximations to the generalized hypergeometric functions. I, J. Math. Pure et Appl., 58 (1979), 445-476. [2] N. I. Fel'dman and Yu. V. Nesterenko, Transcendental Numbers, Number Theory IV, Encyclopaedia of Mathematical Sciences, Vol. 44, Springer.

172


[3] M. Huttner, Problime de Riemann et irration.alite' d'un quotient de deux fonctions hyperge'ome'triques de Gauss, C. R. Acad. Sc. Paris S&ie I, 302 (1986)) 603-606. [4] M. Huttner, Monodromie et approximation diophantienne d'une constante lie'e aux fonctions elliptiques, C. R. Acad. Sc. Paris S6rie I, 304 (1987), 315-318. [5] M. Huttner and T. Matala-aho, Approximations diophantiennes d'une constante like aux fonctions elliptiques, Pub. IRMA, Lille, Vol. 38, XII, 1996, p.19. [6] W. Maier, Potenzreihen irrationalen grenzwertes, J. Reine Angew. Math., 156 (1927)) 93-148. [7] Yu. V. Nesterenko, Hermite-Pad6 approximations of generalized hypergeometric functions, Russian Acad. Sci. Sb. Math., 83 (1995), No.1, 189-219.

THE EVALUATION OF THE SUM OVER ARITHMETIC PROGRESSIONS FOR THE COEFFICIENTS OF THE RANKIN-SELBERG SERIES I1 Yumiko ICHIHARA Graduate School of Mathematics, Nagoya University, Chikvsa-kv, Nagoya, 464-8602, Japan [email protected]

Keywords: Rankin-Selberg series Abstract

We study

C

,sx

c,, where c,s are the coefficients of the Rankin-

n=a(p7')

Selberg- series, p is an odd prime, r is a natural number, and a is also a natural number satisfying (a,p) = 1. For any natural number d, we know the asymptotic formula for C,,, %x(n), where x is a primitive Dirichlet character mod d. This is oLtained by using the VoronoYformula of the Riesz-mean En,, c,x(n)(x - n)2. In particular, in case d = p", the fourth power of the Gauss sum appears in that Voronoiformula. We consider the sum over all characters mod pr, then the fourth power of the Gauss sum produces the hyper-Kloosterman sum. Hence, applying the results of Deligne and Weinstein, we can estimate the error term in the asymptotic formula for C ,I, c,. n=a(p")

1991 Mathematics Subject Classification: llF30.

INTRODUCTION

1.

Let a, d be integers, (a,d) = 1, d 2 1. The author [3] studied the sum C n 1. Here L(s, X ) is the Dirichlet L-function with a Dirichlet character X. This Rankin-Selberg series has the Euler product

and where apand Pp are complex numbers satisfying the following conditions;

and by using Deligne's estimate of the hyper-Kloosterman sum, where xo is the principal character mod p. Some notations used in (1.1) and (1.2) are defined below. And if p3 > x2, we have

In this paper, we generalize the result of [3] to the case d = pr by using Weinstein's estimate and an induction argument. First of all, we introduce the Rankin-Selberg series. Let f (z) and g(z) be normalized Hecke eigen cusp forms of weight k and 1 respectively for SL2(Z), and denote the Fourier expansion of them as

Here bar means the complex conjugate. We find that

and

from (1.1). Deligne's estimate of an and bn gives the estimate c, K n'. We find En,, lc,l

Lemma. Let p be an odd prime, r 2 1 a natural number and b a constant with (b,p) = 1. Then we have where q5 is the Euler function, xo is the principal character mod p, k a ( s ) at s = 1 and is the residue of L where

In particular, the error t e n can be estimated as 0(x3/5p3'/5) in the case r 2 3. Here it is noted that Lf@,(O,x) = 0, if k = 1 and x is not trivial. The constant

cy

C1means the sum over the primitive

characters.

Proof of Lemma. The proof is complete in [3] when r = 1. We prove this Lemma when r >_ 2. First, we consider the case r 2 3. Let b-l be the integer satisfying bb-' = 1 mod pT.

is given by

where the integral runs over a fundamental domain for SL2(Z) in the upper half plane. This constant is given by Rankin [7]. The author would like to express her gratitude to Professor Shigeki Egami for his comment.

2.

THE PROOF OF THEOREM

We prepare several facts for the proof of Theorem. First, recall the following functional equation of L f8g(s, x), which was got by Li (61. Let

then we have

where Cx is a constant depending on x with ICxI = 1. When x is a non-real primitive character mod p' (p is a prime), Li [6] shows Cx =

x2, then we have

and

Here, the integral paths C and Ca,bconform to Hafner's notation. Let R be a real number satisfying R > (k 1)/2 - 1. The path C is the rectangle with vertices b fi R and 1- b fi R and has positive orientation. The path Ca,bis the oriented polygonal path with vertices a-ioo, a-iR, b-iR, b+iR, a + i R and a+ioo. In our case, a = 0 and b > (k+E)/2-1 (see Hafner [2]). F'rom the definition, we see that D,(X) = Dp-1 (x) and there are the analogous relations for Qp(x) and fp(x). Hafner [2] showed the asymptotic expansion of fp(x) for x 2 1. (Actually Hafner stated that it holds for x > 0, but this is a slip.)

+

&

We start explaining the sketch of the proof of Proposition. Voronoi formula (2.2) with p = 2 implies

The

If p3' 1 x2, then we have

where 4 is the Euler function and C' means the sum over primitive characters. Here if k = 1 and x is not trivial, then LfBg(O,X) = 0. The basic structure of the proof of Proposition is same as the argument which is used in the proof of Proposition 1 of the author [3]. (This method is used in Golubeva-Fomenko [I] and the author [3], and the idea goes back to the works of Landau [5] and Walfisz [9].) Therefore we just give a sketch of the proof of Proposition in this paper.

The left-hand side of (2.3) is equal to

Let T be a real number satisfying xE < 7 5 x, and we define the operator AT as

180


The evaluation of the s u m over arithmetic progressions . . . I1

where h ( x ) is a function. We consider the operation of A, to (2.3) and (2.4). Then we get the following result.

-

n-i(pr)

n r a (p')

J

=p

J U T

x

v

(

c

c )

+

%

dwdv

x x 3 ~ - 4 p 4 rin (2.9) by using Hafner's estimate of f2 ( x ). Then this part is estimated as 0 ( r1/2x3/2p3r/2$(pr)). The estimate of the sum over p4'/167r4 < n 5 x 3 ~ - 4 p 4 ris obtained by using (2.10) and Hafner's estimate of f o ( x ) . In fact, we can estimate it The reason of the restriction p4r/16a4 < n is as that Hafner's estimate of f p ( x ) can not use in x < 1. The estimate of the remaining part sum over p4'/16?r4 2 n can be got by using (2.10) and moving the integral path of fo appropriately, similarly to the argument in [3].We are able to estimate this part of (2.9) as o ( T ~ ~ -d)(pT)>. ~ ~ ~ Collecting the above results, we obtain

n=a (p')

In the same way, we have

We put T = x3/5p3'/5, then we complete the proof of the latter half of 1% 1 1) whose conjugates other than itself have modulus less than one. Let Q(P) be the smallest extension field of rational numbers Q containing P. T h e o r e m 0.1 ( B e r t r a n d , K. Schmidt). Let P be a Pisot number and let x be a real number of [0, 1). Then x has an eventually P-expansion if and only if x E Q(P). In [I], Akiyama investigated sufficient condition of pure periodicity where /3 belongs to a certain class of Pisot numbers. Authors [6] characterized numbers having purely periodic P-expansions where P is a Pisot number satisfying the polynomial Irr(P) = x3 - klx2 - kzx - 1,O 5 k2 k1 # 0. In [lo], one of the authors gives necessary and sufficient condition of pure periodicity where P is a Pisot number whose minimal polynomial is given by

=

x (1 -+ ca)

ncN

Since the edge set E(G) is finite, there exists

{f

( ti! )

n= 1

Here each edge e E E(G) starts at a vertex denoted by I(e) E V(G) and terminates at a vertex T ( e )E V(G) . On the plane P we can define the sets Xi (1 5 i 5 d) and X using the edge shift XG as follows:

-+

Moreover, the set X has the following property:

188


Substitutions, atomic surfaces, and periodic beta expansions

189

where f is the lattice given by f = ~ f nis(el = -~ei) I ni E Z } . See the details in [4]. F'rom the Baire-Hausdorff theory) we see that the set X has at least one inner point. Then 1x1is positive, where we note I K I the measure of a set K. In [4], it is implied that X is the closure of the interior of X.

{

Proposition 1.2. For any 1 5 i 5 d, the following set equations hold: In order t o know Xi are disjoint each other up to a set of measure 0, we would prepare lemmas. The next result can be found in [2], originally in [8].

Lemma 1.3. Let M be a primitive matrix with a maximal eigenvalue A. Suppose that v is a positive vector such that M v 2 Xv. Then the inequality is an equality and v is the eigenvector with respect to A.

Proof. The definitions of Xi imply that O 0 - l ~ ~

Lemma 1.4. The vector of volumes inequality:

substituting

( ) (: ) ( for

and

jnl kn-1

) ( ) for

Oa for all n 2 2,

(I"')

>3 lXdl

satisfies the following

(y. lXd

l

Proof. F'rom the form of Xi in the equation (4), we see

(t)

w f ) = i , ~ ( L : ) and =~,

n€N

1

we know that Ioo-'xi Since the determinant of Oo-' restricted to P is 0, [Xi1. Hence we arrive at the conclusion. 0

=

Two lemmas above imply the following result.

We can get the set equations above. Applying Oo to the equation (4), from the form of the substitution a in the equation (3))we have

Corollary 1.5. The sets Xi (1 5 i 5 d ) are disjoint up to a set of measure 0. Therefore the atomic surface has the partition (5).

'

Proof. Lemmas 1.3 and 1.4 show that the vector of volumes (IXi1) l 0 such that T;(x) is reduced. Proposition 3.3 says that T;(x) has a purely periodic P-expansion. Hence x has an eventually periodic P-expansion. The opposite side is trivial. 2. Necessity is obtained by Proposition 3.3. Conversely, assume that x has a purely periodic P-expansion. From 1, we know x E Q(P). According t o Proposition 3.4, there exists N > 0 such that T;(X) is reduced. The pure periodicity of x implies that there exists j > 0 such that 'T : (x) = x. Lemma 3.2 1 says that x is reduced. J

h

By the lemmas above, we can get sufficient condition of pure periodicity of P-expansions.

Proposition 3.3. Let x E Q(P) n [O, 1) be reduced. Then x has a purely periodic P-expansion. Proof. Lemma 3.2 2 shows that there exist xf (i 2 0) such that xf are reduced and Tp(xf) = where we set x; = x. As Y is bounded, we can see the set { x , * )is~ a~ finite set. Hence there exist j and k ( j > k) such that x; = x;-& Hence T/(x) = x. Therefore x has a 0 purely periodic P-expansion. h

Proposition 3.4. Let x E Q(P) n [O,l). Then there exists Nl that are reduced for any N Nl .

TFX

>

> 0 such

--k

Proof. Simple computations show that Sp (p(wx)) exponentially comes as k -+ GO. Since there exist the finite number of p w T/(x)) near

P

--N1

in a certain bounded domain, Sp

(

( ~ ( w x )= ) p (w . T/IN'(x)) E

P for

sufficiently large Nl. Then T? (x) is reduced. From Lemma 3.2 1, we 0 see that T ~ ( X are ) reduced for any N N l .

>

Finally, we can get our Main Theorem.

References [I] S. Akiyama, Pisot numbers and greedy algorithm, Number Theory, Diophantine, Computational and Algebraic Aspects, Edited by K. Gyory, A. Petho and V. T . S6s, 9-21, de Gruyter 1998. [2] P. Arnoux and Sh. Ito, Pisot substitutions and Rauzy fractals, PrBtirage IML 98-18, preprint submitted. [3] A. Bertrand, DBveloppements en base de Pisot et &partition modulo 1, C.R. Acad. Sci, Paris 285 (1977), 419-421. [4] De Jun Feng, M. Furukado, Sh. Ito, and Jun Wu, Pisot substitutions and the Hausdorff dimension of boundaries of Atomic surfaces, preprint . [5] Sh. Ito and H. Ei, Tilings from characteristic polynomials of Pexpansion, preprint . [6] S. Ito and Y. Sano, On periodic P-expansions of Pisot Numbers and Rauzy fractals, Osaka J. Math. 38 (2001), 1-20. [7] W. Parry, On the P-expansions of real numbers, Acta Math. Acad. Sci. Hungar. 11 (1960), 401-416. [8] M. QueffBlec, Substitution Dynamical Systems - Spectral Analysis, Springer - Verlag Lecture Notes in Math. 1294, New York, 1987. [9] A. RBnyi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar. 8 (1957), 477-493. [lo] Y. Sano, On purely periodic P-expansions of Pisot Numbers, preprint submitted.

194


[ll] K. Schmidt, On periodic expansions of Pisot numbers and Salem numbers, Bull. London Math. Soc., 12 (1980), 269-278.

THE LARGEST PRIME FACTOR OF INTEGERS IN THE SHORT INTERVAL Chaohua JIA Institute of Mathematics, Academia Sinica, Beijing, China

Keywords: prime number, sieve method, Buchstab's identity Abstract

In this report, the progress on the largest prime factor of integers in the short interval of two kinds is introduced.

2000 Mathematics Subject Classification 1 lN05.

I. THE DISTRIBUTION OF PRIME NUMBERS In analytic number theory, one of important topics is the distribution of prime numbers. We often study the function

) the zeros of complex function In 1859, Riemann connected ~ ( x with c(s). He put forward his famous hypothesis that all non-trivial zeros of xa.

we employ the Fourier expansion

11. THE LARGEST PRIME FACTOR OF INTEGERS IN THE SHORT INTERVAL (z, x + xi)

+

Firstly we decompose all integers in the interval (x, x x i ) . Among all prime factors, there is the largest one which is denoted as P ( x ) . We have an identity

where

t o get the trigonometric sum

where ((x)) = {x) - and e(x) = e2"ix. Using Vaughan's identity, we get two kinds of trigonometric sum. One is

C

~ a ( m ) ~ b ( n ) e ( E ) , n O. The first result on this topic is due to Jutila. He [18] proved that where 29 = Bolog [4] improved it to 29 = 0.772. Balog, Q ( z ) 2 x"', Harman and Pintz [5] obtained 29 = 0.82. They also used the basic frame in the topic 11. But now they could use the Perron's formula

+

i.

t o transform sums into integrals. Then they applied some mean value formulas to deal with these integrals. Let

We have a formula similar to (3). In the first sum

- k,

&,

fl(nv) = v, [a,P] is the image of where 1 f "(x) 1 1 f "'(x) 1 [a, b] under the transformation y = f '(x) and llxll denotes the nearest distance from x to integers. One can refer to [15]. The reverse formula is based on Poisson's summation. A suitable combination of Weyl's inequality and reverse formula yields good estimate for trigonometric sum. On the new estimate for trigonometric sum, Theorem of Fouvry and Iwaniec [8] plays an important role, which transforms the trigonometric sum into suitable integral. Ramachandra [20] proved that P ( x ) > 1.9, where p = Graham [9] got p = 0.662. He used the sieve method and the Fourier expansion. As the development of the estimate for trigonometric sum and the sieve method, some new exponents have appeared. There are exponents of Jia I141 ( p = 0.69), Baker [I] ( p = 0.70), Jia [15] ( p = 0.728), Baker and Harman [2] ( p = 0.732), Liu and Wu [I91 ( p = 0.738). If we can get p = 1, then we can prove that there is a prime in the short interval (x, x x i ) . But the present results are far from the optimal one. The main reason is that on this topic, we can not use Perron's formula and the mean value estimate for Dirichlet polynomials.

Vaughan's identity, Perron's formula, mean value formulas

and

g.

+

are used. In the second sum

the sieve method is applied. The error term in the sieve method can be dealt with by the estimate of Deshouillers and Iwaniec [6]

200

A N A L Y T I C NUMBER THEORY

The largest prime factor of integers i n the short interval

This estimate which is based on the theory of modular forms is better than the classical one (5). In 1996, Heath-Brown made a great progress on this topic. He [12] got 19 = $. Heath-Brown had an innovation on the basic frame. He considered the sum of the form

where

201

* means some conditions on pl, p2 and p3, one considers the sum

and uses the relationship

-

-

where p T VP, pl PE, , ps P E . If one can prove > 0, then there is a prime factor p P. Hence, the largest prime factor Q(x) > P. In the previous papers, the sum

to get the expression

is considered, where p 5 xa and 1 is an integer. In Heath-Brown's device, the prime factors pl, . . , ps are more flexible than I. Heath-Brown introduced some new ideas on the sieve method. His ideas can be traced back to Linnik's identity. Let

Then in the different ranges of d and 1, one can apply different mean value formulas. In the joint work of Heath-Brown and Jia [13] in 1998, we got 6' = In this paper, we used Heath-Brown's innovation but applied the traditional sieve method. Harman's method was employed (see [lo]). The major feature of Harman's method is that one can get an asymptotic formula for the sum

.-•

We have

g.

where r is a small positive constant, while usually one can only take = E . Harman's method works in some topics since there is the estimate for the sum of type 11. By Harman's method, in the sum (9), we only decompose one of pl, p2, p3 and keep the others unchanged. In this way, one can use the mean value estimate T

Comparing the equations (7) and (8), we can get some identities on the sieve method. Now I give a quite rough explanation on the application of the above identity. For the sum

The estimate (10) is due to Deshouillers and Iwaniec [7], which depcnds on their work for the application of the theory of modular forms. It is difficult to apply formula (10) in Heath-Brown's original work [12]. Moreover, we used computer to deal with the complicated relationship among the sieve functions. By the help of computer, we can get good estimate for Buchstab's function w(u) which is defined as

202

T h e largest prime factor of integers in the short interval


203

[lo] G. Harman, On the distribution of cup modulo one, J . London Math.

We have 0.5607 5 w(u) 5 0.5644, 0.5612 5 w(u) 5 0.5617,

u 2 3, u 2 4.

One could refer to [16]. Before we only had Jingrun Chen's result that w(u) 5 0.5673 for u 2 2. Recently Jia and M.-C. Liu (171 got a new exponent 29 = $. In this paper, we employed Harman's new idea on the sieve method (see [Ill). In some sums of the form

Harman applied the sieve method to the variable p, which is similar to Jingrun Chen's dual principle. In the sum pl p2, Chen applied the sieve method to pl, then to p2. We used the work of Deshouillers and Iwaniec (71 again in more delicate way and made complicated calculation in the sieve method. Then we got the new exponent 29 = $.

+

References [I] R. C. Baker, The greatest prime factor of the integers in an interval, Acta Arith. 4 7 (1986)) 193-231. [2] R. C. Baker and G. Harman, Numbers with a large prime factor, Acta Arith. 7 3 (1995), 119-145. [3] R. C. Baker and G. Harman, The difference between consecutive primes, Proc. London Math. Soc. (3) 72 (1996), 261-280. [4] A. Balog, Numbers with a large prime factor 11, Topics in classical number theory, Coll. Math. Soc. Jdnos Bolyai 34, Elsevier NorthHolland, Amsterdam (l984), 49-67. [5] A. Balog, G. Harman and J. Pintz, Numbers with a large prime factor IV, J . London Math. Soc. (2) 28 (1983), 218-226. 161 J.-M. Deshouillers and H. Iwaniec, Power mean values of the Riemann zeta-function, Mathematika 29 (l982), 202-2 12. [7] J.-M. Deshouillers and H. Iwaniec, Power mean-values for Dirichlet's polynomials and the Riemann zeta-function, 11, Acta Arith. 4 3 (l984), 305-312. [8] E. Fouvry and H. Iwaniec, Exponential sums with monomials, J. Number Theory 33 (1989), 311-333. [9] S. W. Graham, The greatest prime factor of the integers in an interval, J. London Math. Soc. (2) 24 (1981), 427-440.

SOC.(2) 2 7 (1983), 9-18. [ll] G. Harman, On the distribution of cup modulo one 11, Proc. London Math. Soc. (3) 72 (1996), 241-260. [12] D. R. Heath-Brown, The largest prime factor of the integers in an interval, Science in China, Series A, 3 9 (1996), 449-476. [13] D. R. Heath-Brown and C. Jia, The largest prime factor of the integers in an interval, 11, J. Reine Angew. Math. 498 (1998), 3559. [14] C. Jia, The greatest prime factor of the integers in a short interval (I), Acta Math. Sin. 29 (1986), 815-825, in Chinese. [15] C. Jia, The greatest prime factor of the integers in a short interval (IV), Acta Math. Sin., New Series, 1 2 (1996), 433-445. [16] C. Jia, Almost all short intervals containing prime numbers, Acta Arith. 76 (1996), 21-84. [I71 C. Jia and M.-C. Liu, On the largest prime factor of integers, Acta Arith. 9 5 (2000), 17-48. [18] M. Jutila, On numbers with a large prime factor, J . Indian Math. Soc. (N. S.) 3 7 (1973), 43-53. [I91 H. Liu and J . Wu, Numbers with a large prime factor, Acta Arith. 89 (1999)) 163-187. [20] K. Ramachandra, A note on numbers with a large prime factor 11, J . Indian Math. Soc. 3 4 (1970), 39-48. [21] A. Selberg, On the normal density of primes in short intervals, and the difference between consecutive primes, Arch. Math. Naturvid. 4 7 (l943), 87-105.

A GENERAL DIVISOR PROBLEM IN LANDAU'S FRAMEWORK S. KANEMITSU Graduate School of Advanced Technology, University of Kinki, Iizuka, Fzlkuoka 8208555, Japan

A. SANKARANARAYANAN School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai-4 00 005, India

Dedicated to Professor Hari M. Srivastava on his sixtieth birthday

Keywords: divisor problem, mean value theorem, functional equation, zeta-function Abstract

In this paper we shall consider the general divisor problem which arises by raising the generating zeta-fuction Z(s) to the k-th power, where the zeta-functions in question are the most general E. Landau's type ones that satisfy the functional equations with multiple gamma factors. Instead of simply applying Landau's colossal theorem to Z k(s), we start from the functional equation satisfied by Z(s) and raise it to the k-th power. This, together with the strong mean value theorem of H. L. Montgomery and R. Vaughan, and K. Ramachandra's reasonings, enables us to improve earlier results of Landau and K. Chandrasekharan and R. Narasimhan in some range of intervening parameters.

2000 Mathematics Subject Classification: 1lN37, 1lM4l.

1.

INTRODUCTION

In order to treat a general divisor problem for quadratic forms (first investigated by the second author [12]) in a more general setting, we shall work with well-known E. Landau's framework of Dirichlet series satisfying the functional equation with multiple gamma factors where the number of gamma factors may not necessarily be the same on both sides [6], [13]. The main feature is that we raise the generating Dirichlet series Z ( s ) to the k-th power while both Landau [6] and Chandrasekharan and 205

C. Jia and K. Matsumoto (eds.), Analytic Number Theory, 205-221.

206

A general divisor problem i n Landau's framework


Narasimhan [2] simply apply their theory of functional equation to Zk(s) and that we incorporate the mean value theorem in the estimation of the resulting integral, providing herewith some improvements over the results of Landau [6]and Chandrasekharan and Narasimhan [I] in certain ranges of intervening parameters. The main ingredient underlying the second feature is similar to K. Chandrasekharan and R. Narasimhan's approach; but in [2] they appeal to their famous approximate functional equation whereas in this paper we use a substitute for it (Lemma 3.3), following the idea of K. Ramachandra [8]-[ll], we avoid the use of it thus giving a more direct approach to the general divisor problem. To state main results, we shall fix the setting in which we work and some notation. 1. Let {an} and {bn} be two sequences of complex numbers satisfying

207

are gamma factors and where the real parameters ai, Pi, 1 5 i 5 p , yj, Sj, 1 5 j 5 v are subject to the conditions

5. Let

and suppose that q satisfies 1 q > l a n d q > a + - .2

for every

E

> 0, where a 2 0 is a fixed real number.

2. Form the Dirichlet series (s = a

+ it) .-

Z(s) = n= 1

5 and ~ ns

which are absolutely convergent for a

( s=) n=l

5 nS

6. For any fixed integer k

> 2, we define ak(n) by

> a + 1 by Condition 1.

3. We suppose that Z(s) can be continued to a meromorphic function in any finite strip a1 a 5 0 2 such that 0 2 2 a 1 with only real poles, and satisfies the convexity condition there:

0. 4. For a

Also put

so that by (1.2)

Now we are in a position to state the main results of the paper.

Theorem 1. We write

< 0 suppose Z(s) satisfies the functional equation

where A1, A2, A3 are positive numbers and A(s) =

n

r(ai

+ a s ) and

where Mk(x) is the sum of residues of the function real poles of Zk(S) in the strip 0 < a 5 a 1. Then for every E > 0, we have

+

where X1 is as in (3.3).

5Z k(s) at a11 positive

208

Theorem 2. In addition to the conditions in Theorem 1 suppose also that for s o = a 0 it (with fixed a 0 satisfying 112 a 0 5 a + I),

+

0

Acknowledgment. It gives us a great pleasure to thank Professor Yoshio Tanigawa for scrutinizing our paper thoroughly which resulted in this improved version. The authors would also like to thank the referee whose comments helped us to improve the presentation of the paper.

Then for x

3.

> 0 and c > 0, we have

SOME LEMMAS AND A MEAN VALUE THEOREM

Lemma 3.1. .Let Q = Q(E) = a

NOTATION AND PRELIMINARIES

Z(a

+

+

+ 1+E.

This follows from the approximate formula for the discontinuous integral (see Davenport [3]): Let

where qo = qo(j, a ) is a positive constant depending on j and a , and that j X o < qo. Then we have

2.

209



We use complex variables s = a it, w = u iv. E always denotes a small positive constant and ~ 1~ , 2 . ., . denote small positive constants which may depend on e. cl, cz, . . . denote positive absolute constants. The Stirling formula [15] states that in any fixed strip a1 5 a 0 2 as (tl -+ oo we have

uniformly in

-E

+ 1+ e as in (2.4).

+ it) l we have the approximation

(1

into two parts I; ( n 5 Y ) and I; ( n > Y ) :I' = I; + I;. In I; we move the line of integration back to u = - E committing an error of order 0 ( ~ ~ 5 e - ~ 6 ( ~ ~ g ~ ) ~ ) . Thus

z +it)

I' being the sum of I; and I;, we substitute (3.7) into (3.6) to conclude the assert ion. 0 Proof. By the Mellin transform we have, after truncation using Stirling's formula,

Theorem 3. For T 2 To, where To denotes a large positive constant, we have for every E > 0

(i+ l 2 it)

dt

T and 2n substituting the approximation e - T = 1 0(%)in the former and 2n e - T = o ( ( : ) ~ ~ + ~in ) the latter, we conclude that

+

so that by (2.2)

Since By Lemma 3.2, the inner integral

ST

2T

is

l~

(I +

it) 12dt =

/T27

the assertion follows from (3.9)-(3.11).

by (1.1). Thus, substituting this, we conclude that

Proof. We distinguish two cases

+ 1 5 q.

In Case (i), we get

by (1.9)' and in Case (ii), we get

as above.

dt,

0

Lemma 3.4. For the quantities ql and X1 defined by (1.9) and (3.3)) respectively, we have 2X1 L q1 + 2 q .

(ii) a To estimate the mean square of I; and S, we apply Lemma 3.2. Direct use of the estimate (2.2) and then of that lemma yields

+ lii2 + 11;~)

again by (1.9).

214

4.



Choosing

PROOF OF THEOREM 1 AND 2

Proof of Theorem 1. We move the line of integration in the integral appearing in (2.3) to o = By the theorem of residues we obtain

4.

we get nlx

whence we conclude the assertion of Theorem 1. where Mi(x) is the main term which is the sum of residues of the function Zk(S) in the interval < o 5 o 1, and $.Zk (s) at all possible poles of 7 I f h (resp. I,) denotes the horizont a1 (resp. vertical) integrals:

4

215

+

1 0

Proof of Theorem 2 is similar to that of Theorem 1. For simplicity, we assume oo = 112. Indeed, instead of (4.3) we have

Hence instead of (4.4) we have

by (3.4), while Now, instead of (4.4), the assumed inequality j X o < qo implies that the last error term dominates the second one, whence it follows that In the integral in the error term for I, we factor the integrand 121k into 121 k-2 and 1 212to which we apply (3.2) and (3.8) respectively. Then we get xiT(k-2)~l+~-l+~ (4.3)

4

Since the contribution from possible poles in the interval 0 < o 5 t o the main term is ~ ( x),f we may duly write Mk(x) instead of Mi(x). Hence from (4.1)-(4.3) and the above remark we conclude that

Choosing

we get

thereby completing the proof. Now, by Lemma 3.4, the last error term dominates the middle one (apart from an €-factor). Therefore we have

0

SOME APPLICATIONS 5.1. Let Q = Q(y1,. . . ,yl) be a positive definite quadratic form in 1-variables (1 2 2 an integer). Let Z Q ( s ) be the associated zeta-function, 5.

216


A general divisor problem in Landau's framework

summation being extended over all integer I-tuples not all zero. ZQ(s) can be continued meromorphically over the whole plane with its unique simple pole at s = and satisfies the functional equation of type (1.4):

4

so that we may take 7 = 1/2, H = 1, and also a = 0. We remark here that this value 7 does not satisfy the assumption (1.8). So we cannot use Theorem 1. However we use Theorem 5.5 [15] to take Xo = + E . Instead of Theorem 3, we apply the 4-th power moment ( j = 4) with 710 = 1. In this way we can recover Theorem 12.3 of Titchmarsh [15], i.e.

denotes the reciprocal of Q [14].

where d is the discriminant of Q and Hence we may take in Theorem 1,

where

1

Since a, = b, = O(nr-'+"), we may take a = Also we have X1 = 1 - 1 + c l . Hence Theorem 1 gives x C ak(n) = Res zQ(s)-s=;

S

n<x

217

1

- 1, so that 71 = 1 - 1.

ak

is defined as the least exponent such that

with dk(n) denoting the k-fold divisor function due to Piltz (ak(n) = dk(n)). For k 12, we can take j = 12, Xo = 116, 70 = 2 (from a result of D.R. Heath-Brown). The condition j X o 5 70 is satisfied and hence for k 12 which improves the earlier result slightly for we get a k k 12.

>

+ o (xi-$+&) ,

>

6.

which recovers the theorem of the second author [12]. Remark. For I = 2, Kober [5] has proved a mean value theorem slightly better than Theorem 3.

COMPARISON WITH THE RESULTS OF LANDAU AND OF CHANDRASEKHARAN AND NARASIMHAN 6.1. If we apply Landau's theorem [6], we get

5.2. In the special case where

where a is a positive constant, it is known from [16] that ZQ

(

+ it) 47, which is true in view of (1.8). In this case we further (ii) 71 = 2 a 1, i.e. a q - 1 and a 2 q suppose that

> +

>

+

+

>

4.

5.3. If Z(s) = ((s), then we have the most famous functional equation -

( )( s ) =-

9

( ) 1-s

in which case necessarily H 1

s ) ,

> q. Then (6.2) becomes

218



i.e. (6.3). (iii) ql = 2q 1 - H , i.e. H 5 min{2,2(q - a)). In this case we must < H. have q 2 a + Then solving (6.2) in H gives Thus we have proved

+ 3.

Proposition 6.1. If one of the following conditions are satisfied, then our estimate supersedes Landau's bound (6.1):

(iii)

+ + +

4q 2 0 3 4 ( a 1)

4q - 3 and H 4

219

> 0, then we can dispense with the last error term: (x) - Qo(x) = 0 (X f ) + 0 (zq- (log x)'-') . (6.5)

If in addition, a,

A:

-h+2Av2u

-v2

There being essentially no difference between the sequences {n) and {A,) and {p,), we may compare Chandrasekharan and Narasimhan's result above with ours. In our setting we must have

> -32

< H 5 min{2,2(q - a ) ) and q > a + Q .

6.2. Chandrasekharan and Narasimhan [I], [2] considered the pair of that satisfy the Dirichlet series p(s) = C r = l ~A,L Qand $(s) = C r = l functional equation with equal multiple gamma factor A(s) :

$

and

SH q = - = 6A, etc. (1.7)' 2 Since to our situation, A is transformed into kA, our applications are mostly for a, 2 0, and q (> a 1 = 6) can be as big as 6, (6.5) reads

+

with 6 > 0, where N

We are to choose q2 For comparison of the theory of Landau-Walfisz [17] and Bochner-Chandra -sekharan-Narasimhan, cf. [4]. Their theorem (Theorem 4.1 [I]) states that if the functional equation (1.4)' is satisfied as in [I], in particular, a, > 0, 1 v N, A = CL1a, 1 (see (1.6)' below), and the only singularities of the function cp are poles, then

<

SO

that the two error terms be more or less equal:

By (1.2) and the condition on u, we must have u >

6

1 2 k A ~ + 1 >Z ( l + k H ( ~ +1)) . It is enough to choose u so as to satisfy

Thus we choose

(&-1 2

q2 =

+

for q2 2 0 at our choice, where x' = x O(xl-m-h), q = maximum of the real parts of the singularities of cp, r = maximum order of a pole with the real part v , u = ,8 - f with /? satisfying

We are to prove that

1

2kH)

+ k H ( a + 1)'

-

1

SO

that

220

A general divisor problem in Landau's framework


Hence it suffices to prove

As in 6.1, we distinguish two cases and solve (6.7) in H. Then we can immediately prove the following

Proposition 6.2. If either of the following conditions are satisfied, then our estimate supersedes Chandrasekharan and Narasimhan's bound (6.5):

(ii) a 2 9 - 1 and H

> &(&{(k

- 2)17(2a

+ 1) + 2 ( 2 -~ l ) ( a +

111- 1). We note that Case (iii) in Proposition 1 cannot occur in Proposition 2 in view of (1.6)'.

References [I] K. Chandrasekharan and R. Narasimhan, Functional equations with multiple gamma factors and the average order of arithmetical functions, Ann. of Math. (2) 76 (1962), 93-136. [2] K. Chandrasekharan and R. Narasimhan, The approximate functional equation for a class of zeta-functions, Math. Ann. 152 (1963), 30-64. [3] H. Davenport, Multiplicative number theory, Markham, Chicago 1967; second edition revised by H. L. Montgomery. Springer verlag, New York-Berlin 1980. [4] S. Kanemitsu and I. Kiuchi, Functional equation and asymptotic fomnulas, Mem. Fac. Gen. Edu. Yamaguchi Univ., Sci. Ser. 28 (1994)) 9-54. [5] H. Kober, Ein Mittelwert Epsteinscher Zetafunktionen, Proc. London Math. Soc. (2) 42 (1937), 128-141. [6] E. Landau, ~ b e die r Anzahl der Gitterpunkte in gewgen Bereichen, Nachr. Akad. Wiss. Ges. Gottingen (1912), 687-770 = Collected Works, Vol. 5 (l985), 159-239, Thales Verl., Essen. (See also Parts 11-IV) .

221

[7] H. L. Montgomery and R. Vaughan, Hilbert's inequality, J. London Math. Soc. (2) 8 (1974), 73-82. [8] K. Ramachandra, Some problems of analytic number theory-I, Acta Arith, 31 (1976), 313-324. [9] K. Ramachandra, Some remarks on a theorem of Montgomery and Vaughan, J. Number Theory 11 (1979), 465-471. [lo] K. Ramachandra, A simple proof of the mean fourth power estimate for c(: it) and L ( ; it, x ) , Ann. Scuola Norm. Sup. Pisa C1. Sci. (4) 1 (1974), 81-97. [ll] K. Ramachandra, Application of a theorem of Montgomery and Vaughan to the zeta-function, J. London Math. Soc. (2) 10 (1975), 482-486. [12] A. Sankaranarayanan, On a divisor problem related to the Epstein zeta-function, Arch. Math. (Basel) 65 (1995), 303-309. [13] A. Sankaranarayanan, On a theorem of Landau, (unpublished). [14] C. L. Siegel, Lectures on advanced analytic number theory, Tata Institute of Fundamental Research, Bombay 1961, 1981. [15] E. C. Titchmarsh, The theory of the Riemann zeta-function, The Clarendon Press, Oxford 1951, second edition revised by HeathBrown 1986. [16] E. C. Titchmarsh, On Epstein's zeta-function, Proc. London Math. SOC.(2) 36 (1934), 485-500. [n]A. Walfisz, ~ b e rdie summatorische Punktionen einiger Dirichletscher Reihen-11, Acta Arith. 10 (1964), 71-118.

+

+

ON INHOMOGENEOUS DIOPHANTINE APPROXIMATION AND T H E BORWEINS' ALGORITHM, I1 Takao KOMATSU Faculty of Education, Mie University, Mie, 514-8507 Japan [email protected]

Keywords: Inhomogeneous diophantine approximation, Borweins' algorithm, Continued fractions, quasi-periodic representation Abstract

We obtain the values M ( 8 , 4 ) = lim inflql,, (q(((q8 - 411 by using the algorithm by Borwein and Borwein. Some new results for 8 = e l l s (s 1) are evaluated.

>

1991 Mathematics Subject Classification: 11520, 11580.

INTRODUCTION Let 8 be irrational and 4 real. Throughout this paper we shall assume that q8 - 4 is never integral for any integer q. Define the inhomogeneous approximation constant for the pair 8, 4 1.

where 11 . 11 denotes the distance from the nearest integer. If we use the auxiliary functions

then M (8,4) = min (M+(8,d) ,M- (8,4)). Several authors have treated M (8,+) or M + (8,4) by using their own algorithms (See [2], [3], [4], [5] e.g.), but it has been difficult to find the exact values of M ( 8 , 4 ) for the concrete pair of 8 and 4. In [6] the author establishes the relationship between M(O,4) and the algorithm of Nishioka, Shiokawa and Tamura 181. By using this result. 223 C. Jia and K. Matsumoto (eds.), Analytic Number Theory, 223-242.

-

224

On inhomogeneous Diophantine approximation . . . I1


we can evaluate the exact value of M ( 8 , + ) for any pair of 8 and 4 at least when 8 is a quadratic irrational and q5 E Q(8). Furthermore, in [7] the author demonstrates that the exact value of M ( 8 , 4 ) can be calculated even if 8 is a Hurwitzian number, namely its continued fraction expansion has a quasi-periodic form. In this paper we establish the relation between M(8,+) and the Borweins' algorithm, yielding some new results about the typical Hurwitzian numbers 8 = e and 6' = ellS ( s 2). As usual, 8 = [ao;a1, an, . . .] denotes the simple continued fraction of 8, where

Theorem 1. For any irrational 8 and real integral for any integer q, we have

4 so that

225

q8 - $I is never

Remark. As seen in the proof of Theorem 1, the last two values are considered only if C2,-1 > 0 (so, C4n-1 > 0 by Lemmas below); C2n < O (SO,cin< 0).

>

By applying Theorem 1 we can calculate M ( 8 , 4 ) for any concrete pair (8,6). In special we establish the following two theorems.

Theorem 2. For any integer 1 2 2, we have The n-th convergent p,/q, recurrence relations

= [ao;a1 , . . . , a,] of 8 is then given by the

Remark. It is known that this equation holds for 1 = 2 , 3 and 1 = 4 ~ 6 1 [71). , Theorem 3. For any integers 1 2 and s 2 2 with s = 0 (mod I), we have

>

Borwein and Borwein [I] use the algorithm as follows:

$ = do Yn-110,-1 = dn

+ YO, + ^In,

do = 16J , dn = lm-1/8n-1J

(n = 1 , 2 , . . .).

Then, $ is represented by

2. ,

where Di = qi8-pi = (-1)'Oo81 . . . Bi (i 2 0). Put Cn = C:=l(-l)i-ldiqi-l. Then IIcne - $11 = 1 - {cne - 6) = II(-l)"%Dn-iI1 = 7nlDn-11. We can assume that 0 < 6 1/2 without loss of generality. Then $I can be represented as q5 = Czl(-1)'-Idi ~ ~ - 1 -. 6 is also expanded by the Borweins' algorithm as

0, then d,+l = 0. (2) If d, > 0, then

Proof. It is easy to prove. Lemma 2. If Cn + CA = (-l)n-lqn, then Cn+1 + C;+l = If c n CL = (-l)"-' (qn - qn-1) or Cn CA = (-l)"qn_l, then

+

+

Proof. When n = 1, we have (dl + di) + (yl + 7;) = a1 + 81 because 70

dl+~l=--, 60

d ; + 7 ; 1=-7Yo

and a1

+ 81 = -.1

80

226


On inhomogeneous Diophantine approximation

Notice that dl, di and a1 are integers, yl, yi and 191 are non-integral real numbers. Thus, if 81 > yl, then dl + d; = a1 and y1 + yi = el, yielding Cl C; = (dl +d;)qo = q1. If O1 < 71, then dl + d i = a1 - 1 and yl yi = 81 1, yielding C1 C; = ql - go. Assume that Cn C; = (-l)"-lqn and yn 7; = On. Since O < yn "/ < On 7 we have

+

+

+

+

+

dn+, =

+

=0

and

+

d;+' =

+

[a

=O.

+

+

a l , then choose the odd number n ( 2 3) satisfying qn-2 + l, 1 I K 5 qn. Put Kn = K and zn = [(Kn - ~ n - ~ ) / q ~ -SO~ that 1 I Kn - znqn-l I qn-2, then put 1 I zn I a,. If qn-2 - qn-3 zn-l = 0 and Kn-2 = Kn - ~nqn-1 (SO,~ ~ =~ - 2 Otherwise, put Kn-1 = znqn-l - Kn. If K < 0, then choose the even number n ( 2 2) satisfying -qn + 1 5 K I -qn-2. Put Kn = - K and zn = [(Kn - qn-2 l)/qn-ll, SO that 1 5 zn I a,. If n # 2 and 9,-2 - qn-3 I Kn - znqn-l I qn-2 - 1, then zn-l = 0 and Kn-2 = Kn - znqn-l (SO,zn-2 = Otherwise, put Kn-1 = znqn-l - Kn. By repeating these steps we can determine z,, zn-1, . . . , 22. Finally, put zl = K1. For general i < n we have 0 5 zi I ai and

+

+

Hence, Cn+l C;+i = (-l)"-'qn d;+1)qn = (-l)n-lqn and 'Yn+l+ 'Y;+l = yn/en 7;/On = 1. Assume that Cn CA = - qn-l) and yn 7; = On 1. In this case dn+l = [yn/OnJ 1 and d',+l = [?/,/On J 2 1 because yn > 19, > On. Then and

+ >

. . . I1

+

Next, we can obtain {KO) = C ( - l ) i - l z i ~ i - l . If yn+l

+?A+,

= en+1+ 1, then dn+l +d;+l = an+l and

Notice that if zi-1 = ai-1 then zi = 0 (2 I i 5 n). Put

Since zn # 0 is followed by zn- 1 # an- 1,

3.

PROOF OF THEOREM 1 First of all, any integer K can be uniquely expressed as

+

+

+

- 1) = an-3. and Tn-3 = Tn-20n-3 zn-3 < Bn-2)On-3 Hence, by induction, if # ai then Ti < ai. If zi = ai then Ti < ai Oi

+

228

O n inhomogeneous Diophantine approximation


and z-1 have

< ai-1. Therefore, T,8i-l < (ai + 8i)8i-l

= 1. Especially, we

n

0
4, then IKIJIKO- 411 > 1K14 rn (IKl 00). If //KO-411 = 1-(+-{KO}) > 1-4, then ~ K ~ ~ / > K IKl(1-4) ~ - 4 ~ ~ rn (IKI 4. Suppose that K # Cn or di = zi ( 1 I i s - 1 ) and ds # 2, ( 1 I s I n ) . If ds > z,, then +

+

--+

--+

and

where y,*= Ts+lOs= Ts - zs(< 1 ) . Since Qn-2

+ 15 K I

-qn+ 1 5 K 5

qn -qn-2

n : odd; and n : even

{

1 5 CS I (IS -qs+ 1 5 Cs

0 (so, n is odd), by dn-i > zn-1 0 we have -9,-l+l 5 CnA1 0, and Cn-1 CA-1 = -qn-l or -9,-1 qn-2. If K qn-1 - 1, then K ICn-ll, yielding KllKO - 411 > ICn-lIIICn-le - 411. Assume that K < qn-1 - 1. Then zn = 1. Hence,

>

and

+

>

+

>

0 and zn = 1, then Cn-l CA-l = qn-1 or qn-1 - qn-2. By applying the same argument as the case where K > 0 ( n is odd), Cn-l < 0 and z, = 1 above, we have

231

The first example, Theorem 2, shows the case where 6' is one of the typical Hurwitzian numbers, 6' = e .

Proof of Theorem 2. First of all, we shall look at the cases when 1 = 5 and I = 6. When 6' = e = [2; 1,2i, 4 = 115 is represented as

l]zl,

If z,

> d,, then

+

If s 5 n - 3, IKI > lCsl qs-1 yields the result. Let n be odd. If n be even, the proof is similar. When s = n - 2, we can assume that dn-2 > 0, so Cn-2 > 0. Otherwise, there is a positive integer s ( < n - 2) such that = Cn-2. Then, d, > 0 and C, = Cs+l = and for n = 1 , 2 , . . .

When s = n - 1, we can assume that dn-l > 0, so Cn-l this case is reduced to the case s 5 n - 2. Then,

< 0. Otherwise,

When s = n, we can assume that dn > 0, so Cn > 0. Otherwise, this case is reduced to the case s 5 n - 1. Then, K = (zn - dn)qn-l + Cn 2 Cn + Qn-1. 730n-2

SOME APPLICATIONS We shall denote the representation of 4 (0 < 4 < 1) through the expansion of 6' by the Borweins' algorithm by 4 =e (dl, d2, . . . ,dn, . . . ) 4.

1

= 5,

Notice that

with omission of do = 0. The overline means the periodic or quasiperiodic represent ation. For example, Notice also that ~3nlD3nl=

1 a3n+l+e3n+l+q3n-l/~3n

-+

1 l+O+l

1

=

- (n + W)

232


Since

O n inhomogeneous Diophantine approximation

. . . I1

In a similar manner one can find

1 - $ = 415 is represented as one can have

and

arid

For simplicity we put

and

= (-1

+ 401)/5, Ti = (2 - 02)/5 and for n = 1 , 2 , . . .

233

234


Since

one can have

On inh~omogeneousDiophantine approximation . . . I1

In a similar manner one can find

Therefore, we have M ( e , 115) = 1/50.

4 = 116 is represented as

and

and for n = 1,2,. . .

and

235

236

On inhomogeneous Diophantine approximation . . . 11


Since

and yi = (-1

+ 581)/6, yb = ( 3 - 02)/6, for n = 1 , 2 , . . .

Since one can have

and

In a similar manner one can find that

one can have

I

Ci8n+3 D18n+ 2 Id8n+3

6 and

1 - q5 = 516 is represented as

=-

72

( n -, oo)

237

238

On inhomogeneous Diophantine approximation . . . I1


Therefore, we have M ( e , 116) = 1/72. For general 1, when 1 is even, we have 731n-1 = 031n-l/Z ( n -+ w ) , C31n-1 = ~31n-l/l and d31n-1 = a31,-1/1. Thus,

1 1

= lim -q31n- 1 D31n-2 n-ca

1

Proof of Theorem 3. When s r 0 (mod 1) (1 2 2), 4 = 111 is represented -+

1/(21)

1 1 1 = - ' 1' - = 1 21 212 ' and for i = 1 , 2 , . - . -, oo

Since

Next example, Theorem 3, where 0 = ellS with some integer s 2 2, looks like similar but is much more complicated. Continued fraction expansion of ellS (s 2) is given by

>

we have

In other words,

I n a similar manner one finds that and for n = 1,2,. . . -, oo

+

03n-1 = [O;ll (2n 1)s - 1'1, I , . . .] 1 , Odn = [O; (2n 1)s - 1,1,1, (2n 3)s - 1 , . . .]

+

+

+

-+

239

0.

Notice that

I t is, however, not difficult t o see the specific case, where 1 s 2 2 with s r 0 (mod I).

> 2 and

(C6n-5

+ q6n-6) (1 - 76n-5) 1 D6n-6 1

1

-+

212 (n -+00) .

240

On inhomogeneous Diophantine approximation . . .I1


Therefore, M(ellS,111) = 1/(212) if s = 0 (mod 1) (1

Next, 1 - q5 = 1 - 111 is represented as

> 2).

241 0

The other cases can be achieved in similar ways but the situations are more complicated. Conjecture 1. For any integer s ( 2 1) and 1(> 2)

When 1 is even with 1 5 50, it has been checked that M(elIs, 111) = 1/(212). When 1 is odd with 1 5 50, the following table is obtained. {s (mod 1) : ~ ( e ' l " 111) , = 0)

Since

i=l

we have

In a similar manner one finds that

References - 7kn-5)

p6n-61

(1 - 1)2 212

- ( n -, oo) .

[I] J. M. Borwein and P. B. Borwein, On the generating function of the integer part: [ncu+ 71, J . Number Theory 43 (1993), 293-318.

242


+

[2] J. W. S. Cassels, ~ b e lim, r + ,, x ( 8 x cu - yl, Math. Ann. 127 (1954), 288-304. [3] T . W. Cusick, A. M. Rockett and P. Sziisz, On inhomogeneous Diophantine approximation, J. Number Theory, 48 (1994), 259-283. [4] R. Descombes, Sur la ripartition des sommets d i n e ligne polygonale rigulidre non fermie, Ann. Sci. &ole Norm Sup., 73 (1956), 283-355. [5] T . Komatsu, On inhomogeneous continued fraction expansion and inhomogeneous Diophantine approximation, J. Number Theory, 62 (1997), 192-212. [6] T . Komatsu, On inhomogeneous Diophantine approximation and the Nishioka-Shiokawa-Tamura algorithm, Acta Arith. 86 (1998), 305-324. [7] T . Komatsu, On inhomogeneous Diophantine approximation with some quasi-periodic expressions, Acta Math. Hung. 85 (1999), 311330. [8] K. Nishioka, I. Shiokawa and J. Tamura, Arithmetical properties of a certain power series, J. Number Theory, 42 (1992), 61-87.

ASYMPTOTIC EXPANSIONS OF DOUBLE GAMMA-FUNCTIONS AND RELATED REMARKS Kohji MATSUMOTO Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya 464-8602, Japan

Keywords: double gamma-function, double zeta-function, asymptotic expansion, real quadratic field Abstract

Let I'2(p, (wl, w2)) be the double gamma-function. We prove asymptotic expansions of log r 2 ( P , (1, w)) with respect to w, both when Iwl + +oo and when Iwl -+ 0. Our proof is based on the results on Barnes' double zeta-functions given in the author's former article [12]. We also n I , En)), log p2 (En prove asymptotic expansions of log r2(2cn - I , ( ~ 1, &n) and log p2 (en,E: - E,), where cn is the fundamental unit of K = Q(J4n2 8 n 3). Combining those results with F'ujii's formula [6] [7], we obtain an expansion formula for ("(1; vl), where C(s; vl) is Hecke's zeta-function associated with K.

+ +

1991 Mathematics Subject Classification: llM41, llM99, llR42, 33B99.

1.

INTRODUCTION

This is a continuation of the author's article [12]. We first recall Theorem 1 and its corollaries in [12]. Let p > 0, and w is a non-zero complex number with 1 arg wl < T. The Barnes double zeta-function is defined by

This series is convergent absolutely for Ru > 2, and can be continued meromorphically to the whole u-plane, holomorphic except for the poles a t u = 1 and u = 2. Let C(u), C(u, P) be the Riemann zeta and the Hurwitz zeta-function, respectively, C the complex number field, a fixed number satisfying 243 C. Jia and K. Matsumoto (edr.), Analytic Number Theory, 243-268.

244 0


Asymptotic expansions of double gamma-functions and related remarks

< 00 < T, and put

Theorem 1. For any positive integer N

245

> 2, we have

W , = {w E C I lwl L 1, Iargwl 500) and

Define

(3={1

v(v - 1) . (v - n

+ l)/n!

if n is a positive integer, if n = 0.

Theorem 1 in [12] asserts that for any positive integer N we have

in the region Xv

for w E W,, where the implied constant depends only on N, P and 00. Also we have

> -N + 1 and w E W,, and also

for w E Wo, where Cl(v, P) = 1 - N w E Wo.From (2.10) it follows that

hence

which is again uniform in ,B if 0 the fact

< P 5 1. Noting this uniformity and

where

+

E,

250

Asymptotic expansions of double gamma-functions and related remarks


3.

It is clear that Bo(O;0 ) = (i(0, P) and

251

ADDITIONAL REMARKS ON THEOREM 1 In this supplementary section we give two additional remarks. First we mention an alternative proof of (1.6). Shintani [15] proved

Also, since v

v-0

=-$@)--~-l, 00

we see that

('' ~ F ( I nw) + exp n=l +

lim Bl (v; p) = $(P)

v-0

()'

+ P-'.

{2nw H+ (1 - a ) log(nw)}

(3.1)

(see also Katayama-Ohtsuki [9], p.179). Shintani assumed that w > 0, but (3.1) holds for any complex w with 1 arg w) < T by analytic continuation. We recall Stirling's formula of the form

Hence from (2.12) (with (2.4)) we get

given in p.278, Section 13.6 of Whittaker-Watson [18], where BL+2(a) is the derivative of the (m 2)t h Bernoulli polynomial and M is any positive integer. Noting

+

for N 2 2. The estimate

(

can be shown similarly to (2.8); this time, instead of Lemma 2 of [12], we use the fact that (1 (v, p) and (i (v, ,8) are uniformly bounded with respect to p in the domain of absolute convergence. Hence (2..14) is uniform for any p > 0. From (2.13), (2.14) and this uniformity, we obtain 1 2

= - logw

3 4

- - log2n - (((-1)

(p.267, Section 13.14 of [18]), we obtain log

(fi

('' n= 1 r ( l

nw) exp {@ + (1 - P) log(nw)}) + nw) 2nw +

1 + (I(-l)}w-I + -yw 12 From (3.3) and the fact B3(a) = a3 - (3/2)a2

>

for N 2, w E Wo. From (2.13), (2.14) and (2.15), the assertion (1.7) follows.

+ ( 1 1 2 ) ~it follows that

252

Asymptotic expansions of double gamma-finctions and related remarks


Hence the coefficient of the term of order w-l on the right-hand side of (3.4) vanishes, and so the right-hand side of (3.4) is equal to

that the implied constant in (1.6) does not depend on P if 0 < P 5 1. For general P, it is possible to separate the parts depending on ,f3 from the error term on the right-hand side of (1.6). An application can be found in [ll]. We write B = A + where A is a non-negative integer and 0 < 1. Then we have

P,

Substituting this into the right-hand side of (3.1), and noting (3.5), we arrive at the formula (1.6). Next we discuss a connection with the Dedekind eta-function

n

253

Theorem 2. For any positive integer N and Rv > -N

p
2, and by analytic continuation for X v > 1 - N

+ E.

Hence

Applying (2.4) to the right-hand side, we have

+ 0 ( c y N log C). Proof. For X v

Taking the limit a obtain

-+

0, and using (7.12) and (7.17) with

P

= 1, we

> 2, we have

266


Asymptotic expansions of double gamma-finctions and related remarks

which is, again using the Mellin-Barnes integral formula,

References

That is,

and this identity is valid for Hence

Wv > 1 - N + E

by analytic continuation.

Therefore using (7.10), (7.11) (with cu = 1, ,O = 2) and (8.5) we obtain the assertion of Proposition 3. The error estimate O ( C - log ~ J) can be shown similarly to the remark just after the statement of Proposition 2. Now we can easily complete the proof of Theorem 3, by combining Propositions 1, 2 and 3. Since

(valid at first for Wv > 2 but also valid for any v by analytic continuation), by using (6.4) we find

Also, (7.14) implies that C-l(l; (1,2)) = 0 and C-1 (k; ( l , 2 ) ) = k/12 for k 2. Noting these facts, we can deduce the assertion of Theorem 3 straightforwardly.

>

It should be remarked finally that if we only want to prove Theorem 3, we can shorten the way; in fact, since the left-hand side of (5.10) is equal to

the formulas (6.11), (6.6), (2.5), (2.8), (8.8), (7.10), (7.11) are sufficient to deduce the conclusion of Theorem 3. However the formulas of Propositions 1, 2 and 3 themselves are of interest, therefore we have chosen the above longer but more informative route.

267

: :

[I] T . Arakawa, Generalized eta-functions and certain ray class invariants of real quadratic fields, Math. Ann. 260 (1982), 475-494. [2] T . Arakawa, Dirichlet series C= :l -, Dedekind sums, and Hecke L-functions for real quadratic fields, Comment. Math. Univ. St. Pauli 37 (1988), 209-235. [3] E. W. Barnes, The genesis of the double gamma functions, Proc. London Math. Soc. 31 (1899), 358-381. [4] E. W. Barnes, The theory of the double gamma function, Philos. Trans. Roy. Soc. (A)196 (1901), 265-387. [5] J. Billingham and A. C. King, Uniform asymptotic expansions for the Barnes double gamma function, Proc. Roy. Soc. London Ser. A 453 (1997), 1817-1829. [6] A. Fujii, Some problems of Diophantine approximation and a Kronecker limit formula, in "Investigations in Number Theory", T . Kubota (ed.), Adv. Stud. Pure Math. 13, Kinokuniya, 1988, pp.215-236. [7] A. Fujii, Diophantine approximation, Kronecker's limit formula and the Riemann hypothesis, in "Theorie des Nombres/Number Theory", J.-M. de Koninck and C. Levesque (eds.), Walter de Gruyter, 1989, pp.240-250. [8] E. Hecke, ~ b e analytische r Funktionen und die Verteilung von Zahlen mod. eins, Abh. Math. Sem. Hamburg. Univ. 1 (1921)) 54-76 (= Werke, 313-335). [9] K. Katayama and M. Ohtsuki, On the multiple gamma-functions, Tokyo J. Math. 21 (1998), 159-182. [lo] K. Matsumoto, Asymptotic series for double zeta, double gamma, and Hecke L-functions, Math. Proc. Cambridge Phil. Soc. 123 (1998)) 385-405. [I11 K. Matsumoto, Corrigendum and addendum to "Asymptotic series for double zeta, double gamma, and Hecke L-functions", Math. Proc. Cambridge Phil. Soc., to appear. [12] K. Matsumoto, Asymptotic expansions of double zeta-functions of Barnes, of Shintani, and Eisenstein series, preprint . [13] T . Shintani, On evaluation of zeta functions of totally real algebraic number fields at non-positive integers, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 23 (1976), 393-417. [14] T . Shintani, On a Kronecker limit formula for real quadratic fields, ibid. 24 (1977), 167-199.

268


[15] T . Shintani, A proof of the classical Kronecker limit formula, Tokyo J. Math. 3 (1980), 191-199. [16] E. C. Titchmarsh, "The Theory of the Riemann Zeta-Wnction", Oxford, 1951. [17] M.-F. Vignbras, L'equation fonctionelle de la fonction d t a de Selberg du groupe modulaire PSL(2, Z), in "Journhes Arithmetiques de Luminy, l978", Asthrisque 61, Soc. Math. France, 1979, pp.235249. [18] E. T. Whittaker and G. N. Watson, "A Course of Modern Analysis", 4th ed., Cambridge Univ. Press, 1927.

A NOTE ON A CERTAIN AVERAGE O F L ( ; it,X )

+

Leo MURATA Department of Mathematics, Meijigakuin University, Shirokanedai 1-2-37, Mznato-ku, Tokyo, 108-8636) Japan [email protected]

Keywords: Character sum, q-estimate of L ( i Abstract

+ it, X)

We consider an average of the character sum S(X;0, N) = x(n) (Proposition), and making use of this result, we obtain some averagetype results on the q-estimate of L ( i it, x).

+

1991 Mathematics Subject Classification: llL40, llM06.

INTRODUCTION

1.

Let p be an odd prime, q be a positive integer, x be a Dirichlet character mod q. We define the sums

+

and we are interested in q-estimate of L(+ it, x), i.e. we want to find + it, x ) J as a function in q, and we are not a good upper bound for concerned with its t-aspect. In the long history of the study of (S(X;M, N ) J and + it, x)(, the first important estimate is Pdya-Vinogradov inequality ([I31 [l4], 19l8),

IL($

(~(4

and, by making use of partial summation, this gives, for any 1 L(2

+ it, X ) 0,

270

In 1962, Burgess ([I]) proved the famous estimate; for any natural number r and any positive E, S(X;M, N )

0) S ( x ; M, N ) 1 and x

> 0,

and for any n E Z+we have for z E @ \ (-m,O], and hence by analytic continuation

Applying Lemma 3.1 we then get (3.8). So far we have completed the proof of Theorem 3.2.

Proof. Let A = { ( l , a l , n l ) , . . ., ( l , a k , n k ) , ( - m , 0 , 1 ) } . i) Clearly A = {a, (n,))t,l forms an exact m-cover if and only if A -. 0. If A -. 0, then l l n , - m = 0 by Remark 3.2. (That ~ f l l n=, =~m for any exact m-cover (1.1) is actually a well-known result, it can be found in [P2].) By the equivalence of (3.8) and (3.9), A -. 0 if and only if

zfzl

Remark 3.2. In the case m = 1, that (3.8) implies (3.10) was first realized by the author [Sul] in 1989 and later refound by Porubskf [P4] in 1994. If (3.8) holds, then the formula in (3.10) in the case m = 1 and x = [nl, . . ,nk] yields the equality ~ f == 0.~ In 1989 the author [Sul] obtained Theorem 1.1 and noted that f (x, y ) = $ cot s x over (@ \ Z) x @* is a uniform map into (C, thus for any exact 1-cover (1.1) we have

2

k

1 z - cot (s-)

+ as

k

= c o t s z and

+

1 z as - csc2 (sT)

= csc2(sz) ns s=1 n s s= 1 n: for all z E @ \ Z, this was also given by Beebee [Bell in 1991.

-

l / n s = m, (A) reduces to the latter equality Under the condition in (3.13). So, A 0 if and only if (3.13) holds. N

298


By Theorem 3.2, A

-0

On covering equivalence

if and only if for all x E Z we have

299

Let A = {(A,,as,ns)}~=lE S(C) and z E C. For

we have

+

Any integer x can be written in the form a q N where a E R(n) and q E Z,thus the last equality holds for all x E Z if and only if (3.14) is valid. This ends the proof of part (i). ii) As wA(x) 2 m for all x E Z, wd(x) 2 0 for any x E Z. Obviously A {(wd(r), r, ~)}:=jl. When s > 1 and x > 0, by Theorem 3.2

-

where Bn(x) is the Bernoulli polynomial of degree n. So, A only if k

(F)

xAsnr1~, s=1 therefore

9)

since 0.

(2n - I)! +00 1 >O ( j x)2n (rrN)2" 3=-00

.C +

forallx~W.

= 0 for all n = 0 , 1 , 2 , . . .

-

0 if and

.

For the system B = { ( l , a l , n l ) , . - . , ( l , a k , n k ) ,(-m,O, 1)}, this was proved by A. S. Fraenkel [Fl], [F2] in the case m = 1 and z = 0, by Beebee [Be21 in the case m = 1, and by Porubskf [P2] in the case m € Z+ and z = 0. See also Porubskf [Pl], [P3] and Zniim [Zl] for the case z = 0 with the weights 1 in B replaced by real weights. In 1994 Porubskf [P4] essentially established the above general result. However, before the works of Beebee [Be21 and Porubskf [P4], in 1989 the author [Sul] proved Theorem 1.1 and observed that the function bn(x, y) = yn-' Bn(x) is a uniform map into C for each n E N. In 1988 D. H. Lehmer [Le] showed that Bn(x) is the only monic polynomial of degree n such that

As in the last paragraph, now we have

+

k

x as C ~ o t 2 ~ (- T1, n s ) s=l

For any n E N clearly - m ~ o t ~ ~ - ~ ( 2x O, l )for any x E W.

Clearly this is equivalent to (3.16). We are done. Remark 3.3. (3.16) in the case n = 1 gives the following inequality: 1 l<s n).

Hence, noting Repeating the argument, we get the lemma.

rn

Lemma 2. Let there exist a partition (2) for a matrix A E M T ( s ;Z) with 1 < c < CQ, x E I' such that A-nx E L, for all n E N. Then r > {ACmx;m E Z). Proof. We put x, := Anx (n E Z). Lemma 1 implies r 2 {x,,; so that A ~ I3' {xnncj; m E N) for all 0 2 j < C.

m E N),

(3)

U

(Emp) e j ( A " L , ) ~= L, e=, L, Osn 1. It suffices to show the lemma for t = 2. Changing the basis of ZS as a Z-module if necessary, we may assume that T is an "upper triangular" matrix. Then, we can write

By induction, one can show

holds for all n E Z, where Xij(k) are numbers independent of n , belonging to the splitting field K := Q ( a l , . . . ,a t ) of F(x). Then, for 5 = T (XI,...,xt) E Z t \ {o),

312

Certain words, tilings, their non-periodicity, and substitutions o f . . .


for all n

2 0.

313

Therefore, we obtain

We may suppose y l = xi,, . . . ,y, = xi, form a basis of the space ( s o ,X I , .. .). Extending the basis to a basis of Qt, we may assume that

where V := n l 1. As we have already seen in Lemma 5 that the word W(A;c) with c < w can be obtained from W(A; m ) = (ind A(x)),Ez. by taking residues with mod c. For the computation of the values ind ~ ( x )it, is convenient, in some cases, to consider the Hermitian canonial form Hn = H(dnAdn) for dnA-n with d = det A. Here, the Hermitian canonical form H = H(A) for a matrix A E Mr ( s ;Z) is defined to be a matrix H, that is uniquely determined by

which can be obtained by elementary transformations by multiplying A by unimodular matrices from the left. We put

=

Then Cn(A, U)

-

A-nU (n E Z), so that

by which we can get the value ind u - ~ A u ( x ) .In some cases, for A $! C, we can conclude that A E V by considering Cn(A, E). For instance, (vi) A = [0,4, -2//2,2, -2// induction, we can show

- 1, -2,2].

A(A) = {-2,3 f fi}. By

330



C2n+1(A, E ) =2-2n [I,2n - 1,0//0, an, O//O, 0,2"+']. Hence, setting U

=.

t'

if Tx = (x, y, Z) = (2,1, I ) , (1,2,2). We remark that 'a holds for T = (-1,1, I ) , (1,-1, I ) , (1,1, -1) for c 2 3.

[I,1,0//0,1,0//0,0,1], we obtain

331

#

a

It is remarkable that Example (vii) gives infinitely many matrices A(n) with nontrivial one parameter n such that W (A(n); c) is independent of n. We can make some variants of such an example. For instance, the adjugate matrix Bn of H2n({bm}mEz)belongs to C(3; 4; 3) if {bm}rnEz is a linear recurrence sequence having x3 - x2 - x - 1 as its characteristic polynomial with bo = b2 = 0, bl = 1.

which implies A E D(2; 4,2,2; 3) In the following examples, Cn indicates the matrix Cn(A; E). We put

4.

EXAMPLES (SINGULAR CASE)

In this section, we suppose A E M (s; Z), det A = 0. We consider partitions (2) with s > 1 together with their variants

(vii) An = Hsn({~rn)rnEz)E C(3; 3; 3) holds for all n E Z, where {arn),,z is a linear recurrence sequence having x3 - x2 - 1 as its characteristic polynomial determined by a0 = a1 = 1, a2 = 0. In fact, using An+1 = ULAn = AnUR = (UR = U4, UL = T ~ R )we , get

If c = m, or s = 1 then there does not exist a partition (2), nor (15) for any singular matrix A. Thus, we assume c < CCI in this section. We denote by TA the endomorphism on ZS induced by A E M ( s ; Z) as a Z-module. Then, we easily get the following Remark 5. If there exists a partition (2), then

where

ker TAC A~-'I' U {o}, (ZS\ Im TA) c I' holds; if there exists a partition (IS), then

--

-

-

Cl, A i 2 C2 for a11 n E Z. By the similar which implies A;' manner, we get A: 3C1 for all n E Z, so that A: 3C1An = 3ClAoU;t, which together with 3C1Ao = [3,3,6//3,0,3//0,3,6] 3E, so that An E C(3; 3; 3). Hence, using A;' Cl and Ac2 a(U-'AU;c) Over K ~ : C2, we get a

-

--

holds. In particular, ker TAc Im TA follows. (i) Let A = (bi,j-l)l ZS \ Im TA,we get I' > So by Remark 5, so that

there exists a partition (15) if and only if c = 2, and such a partition is uniquely determined:

A ~ I>' E j for all 0 5 i < c holds. If c = s, then (16) follows. So, we may suppose c < s. Let us assume that an element x E 8, does not belong to I'. Then x E A i r for an integer 0 < i < c, so that there exists a y E F such that A2y = x E 2,. Hence, setting y = T(yl,. ,y,), we get yS-,+i # 0, yS-,+i+l = . = y, = 0, so that y E Ec-i. Hence, y E EC-i C Ac-iI' with some 0 < i < c follows from (17), which contradicts y E r . Therefore, in the case c < s, we must have I' > Ec. Now, suppose s - c = i with 0 < i < c. Then Ai-'I' > Sc+i-l= -,-I = (Z\{O)) x {O)'-' follows from r > E,, SO that A i r 3 o, consequently we can not have (2). Thus, we get s - c = i 2 c, ie., s 2 2c. If s = 2c holds, then we get (16). Suppose s > 2c. Then, we can show r > E2,, and then a contradiction by setting i = s - 2c with 0 < i < c in the same way as above. Thus, we get s = 3c, or s > 3c. If s = 3c, then (16) follows. Repeating the argument, we can arrive a t (i).

-

(ii) Let A = (aidi,j-l)l 1 for all 1 2 i 2 s - 1 (s 2 2). Then there exists a partition (15) if and only if c = 2, and such a partition is uniquely determined: -

r = (Zs-'

1

=

x (Z \ {0}))

u

i

U (Zi-'

x (Z \ aiZ) x zS-')

Proof. x := T(l,0,. . . ,0) @ Im TA,SO that x E I' follows from Remark 5, and so, A x = A2x = A3x = . . = o, which implies c 2 2. Suppose that (15) holds with c = 2. Since any element of X := (ZS-I x (Z\{O}))u (Ulii<s(Zi-l x (Z \ aiZ) x ZS-')) can not be an element of Im TA,we get I' > X by Remark 5. Hence, we obtain

> AX = Y := a l Z x

NON-C-PERIODICITY OF WORDS AND TESSELLATIONS

We give some new definitions of non-periodicity for words and sets of dimension s as follows in a general situation. In this section, s 2 1 denotes a fixed integer. We denote by RS, the Euclidean space with the norm 11 * 11 induced by the usual inner product (*, *). For a set S c RS, we denote by (resp., So) the closure (resp., the interior) of S with respect to the usual topology. We mean by a cone J a closed convex set satisfying the following conditions: (i) JO # 4, (ii) If x E J , then rx E J for all r E W+, where R+ is the set of non-negative numbers. We denote by Q, the set of all cones in RS. Note that RS, W,: a half-space in Rs are cones E P,, and that any cone J E Pt, J c Rt x {o)'-' c Rs (t < s ) can not be an element of P,. Any cone J becomes a monoid with o as its unit with respect to the addition, so that J > x S holds for any x E J , S c J. We say a set S c WS is spreading if for any bounded B c S. set B c WS, there exists an element x E RS such that x For instance, Untl B(log(1 + n); T(n2,n3,. . . ,nS+')) is a spreading set, where B(r, a) denotes the open ball {x E RS; [la - XI[ < r}; and so is a set x + J for all x E WS, J E Qs. In what follows, C, denotes the set of all spreading sets in WS. We denote by L the fixed lattice L = L ( P ) := P(ZS) ( P E GL(s; W)) as in Section 2. We say a subset S of WS is spreading with respect to L if for any bounded set B C L, there exists an element x E RS such that x + B c S. C(L) denotes the set of all spreading set with respect to L. For instance, X n L is spreading with respect to L for any X E C,. First, we give definitions for non-periodicity for words W = (w,) E K~ (K # 4) on a set X c L. Note that K can be an infinite set. We denote by Wls its restriction to S:

+

l$i<s A r = a l Z x . . . x a,-lZ x {O).

AI'

5.

x aS-lZ x {o).

Note that s 2 2 implies AX 3 o. Noting X U Y = ZS is a disjoint union, we get r = X and AI'= Y. We can prove the following (iii) by almost the same fashion as (ii).

+

P

334



which is a word on S n X . For words W = ( w ~ )V ~= (~ ~~ 1y, , ~we) ~ ~ write W = V if they coincide as maps; and write W = V if there exists t E L such that W = V t as in Section 2. We say a word W E K X ( X c L) is non-@-periodic if

We say that a word W E K X ( X E L) is non-C-periodic if

Note that, in general, Wls = WIT with S C X implies T C X for W E K~ ( X c L). We say a word is *-periodic (resp., C-periodic) if it is not non-9-periodic (resp., not non-C-periodic). Note that, by the definition, any word on X C L is C-periodic (resp., 9-periodic) if X &f C(L) (resp., if X > p J n L does not hold for all p E L, J E 9,). We remark that both definitions of non-periodicity given above are considerably strong. In fact, the non-fD-periodicity excludes some trivially "non-periodic" words on J E 9, for all s 2 2; for s = 1, the definition requires the non-periodicity for both directions for words on Z, while the definition is equivalent to the usual one for words on N. In general, the non-C-periodicity implies the non-9- periodicity. Note that some non-*-periodic words are C-periodic. For instance, any non-fD -periodic word over K # 4 on N of the form

+

u1va1u2va2 . . unvan .

.

( lim an = 12-00

00;

i

',

W ([I, -1/11, 11; c) for even c follows from Theorem 5 below, cf. the example (ii), Section 3. Secondly, we give definitions for non-periodicity for tilings. A set 8 c WS is called a tile if 8 is an arcwise connected closed set such that O0 = 8. We say that a set 0 C WS is a tessera if 8 is a compact tile. We say that 8 = {O,; p E M ) is a tiling (resp., a tessellation) of J ( J E fD,) if all the sets 0, ( p E M) are tiles (resp., tesserae) such that UpEMea = J and 0; n 0; = 4 (p # Y ) hold. We say that a set 8 = 18,; p E M} (0, c Rs) is C-distributed on a set X c RS if X \ (uPEMOP)&f CS. A set 8 = {O,; p E M} will be referred to as a mosaic on X E Cs if 0, C X are relatively closed sets with respect to X such that 8; n 8; = 4 ( p # v), and 8 is C-distributed on X E C,. Any tiling of J is a mosaic on J . Let 8 = {O,; p E M ) be a mosaic on X . We denote by @Is the S-restriction:

Note that for any mosaic 8, its restriction elyis always a mosaic for a spreading set Y c X. For two sets 8 = {O,; p E M ) , B = {py; v E N) (O,, p,, c WS), we write

if there exists a z E RS such that

u,, v E K* \ {A})

is C-periodic. In particular, any word coming from a normal number is non-9-periodic, but it is C-periodic. On the other hand, for instance, the Thue-Morse word and the Fibonacci word are non-C-periodic, cf. Remarks 6-7. Remark 6. A word W over K # 4 on N is non-C-periodic if there exists an integer m > 1 such that W is m-th power free ( i e . , W has no subwords of the form urn, u E K* \ {A)). In general, any non-periodic (in the usual sense) fixed point of a primitive substitution is m-th power free for all sufficiently large m (cf. [I]), so that it is non-C-periodic. Remark 7. All the Sturmian words ( 2 . e., the words having the complexity p(n) = n + 1) on N are non-Cperiodic. In particular, the Fibonacci word is non-C-periodic.

A

$&

335

We can show that the words W(A; c) are non-C-periodic for some E (Bdd), c > 1. For instance, the non-Cperiodicity of the word

We say that a mosaic {O,;

p E M ) on X is non-@-periodic if

We say that a mosaic {O,;

p E M } on X is non-C-periodic if

Note that any mosaic {O,; p E M ) such that one of the sets 0, is a spreading set is C-periodic, which can be easily seen. Finally, we give a definition of non-periodicity for a discrete set S = E M } c RS. We say that S is a mosaic on X if so is the set {{I,}; p E M ) ; and that S is non-Wperiodic (resp., non-C-periodic) if

{x,; p

-

336

Certain words, tilings, their non-periodicity, and substitutions of.


so is the mosaic { { x ~ ) ; p E M). For instance, A ( P ) is a Q-periodic mosaic for any A E GL(s; R) ; Z U a!Z is a non-Q-periodic mosaic for any a! E R \ Q; while (Z U a Z ) x Z is a Q-periodic mosaic. For a discrete set X c RS, the Voronoi cell T ( X , x ) with respect to x E X is defined by T ( X , x) := {y E RS; llx - yll

2 Ily - xll

for all x E X

337

j < c). We Suppose 1 < c < oo. Choose any xo E A~I'(A,c) (0 shall show that Y(A~I'(A,c), s o ) is a compact set. By Corollary 2, we have xo E AjI'(A, C) = Uo5m..m TCm+j(A),SO that xo E Tcm+j(A), i. e., ind A(XO)= crn j for some m = m(xo) 2 0, and 0 2 j < c. On the other hand, from the definition of ind A it follows

+

\ {x}}.

For A E (P,(c)), we denote by r ( A , c) the set determined by (2). Since any Voronoi cell T is a finite intersection of closed half-spaces, T is a closed convex set, so that it is arcwise connected. Hence, T (AjI'(A, c) ,x ) is a tile for any A E (P,(c)), 0 2 j < c, and x E AjI'(A,c), so that the sets R(A, c; j ) := {T(Ajl?(A,c), x ) ; x E A~I'(A,c)} (0 2 j

< c)

Hence, noting that ind A(+) > cm

Theorem 4. Let A E (Bdd), 0 5 j < c, 1 < c 2 oo. Then the set R(A,c; j ) is always a tessellation consisting of tesserae which are polyhedra of dimension s. For 1 < c < oo, all the tessellations R(A, c; j ) (0 j j < c) are non-C-periodic if and only if so is the word W ( A ;c). The word W(A; oo) is non-C-periodic; while, R(A, oo; j) ( j 2 0) are Q-periodic. Proof. We show that R(A, c; j ) is always a tessellation for A E (Bdd). We suppose A E (Bdd), so that det A # 0. Then, Aj(ZS) becomes a free Z-module of rank s. In fact, the elements

+ j for any x E ACm+j+'(ZS),we get

Thus, setting

F := {XO

become tilings of RS. Note that R(A, c; j ) is not always a tessellation. In fact, some of the cells T (AjI'(A, c), x ) are unbounded, cf. the examples (i-iii) for the singular case in Section 4. We can show the following

+ {fbl, . . . , fbs}} U {so} with bi := b g + j + l (1 5 i 2 s),

we obtain

F c A ~ ~ ( c), A, so that T ( F , XO)3 T ( A 3 1 ' ( ~c), , xo). In addition, T (Ajr(A, c), xo) is a closed set by the definition of Voronoi cells. Hence, it suffices to show that the set T ( F , xo) is bounded for the proof of the compactness of T (AJI'(A,c), xo). For simplicity, we consider T (-xo + F, o ) that is congruent to T (F,xo). By the definition of a Voronoi cell, we have

where F, := {&2-' bl, . . . ,f2-I b,}, and H-(b) denotes a closed half space H-(b) := {x E RS; (b, x) 5 0}, b E Rs \ {o}.

form a basis of Aj (ZS) as a Z-module, i. e.,

where ei is the i-th fundamental vector. The vectors bj( 2 ) are linearly independent over R, so that RS \ Aj (ZS) is not a spreading set. Hence, Aj(ZS) is a mosaic. In view of Corollary 2, we have Tj(A) = AjI'(A, oo), which is not empty by Theorem 1. Since

Since bi (1 2 i 2 s) are linearly independent over R, by the orthonormalization of Schmidt, we can take an orthonormal basis of RS as a vector space over R such that we can write

with respect to the basis. Then, for any E = T ( ~ l , E , ~E) { l , - l J S , there exists a unique element V(E)= (vl (E), . . ,us (E)) E RS satisfying the equations:

we get Tj(A) = A3r(A,00) = A' (ZS)\ A'+' (ZS) # 4, 0

..

=< j

< GO.

(19)

338

R $


I

+

Thus, we have shown that the set R(A, c; j ) is always a tessellation for 1 < c < oo. Note that, since Aj(r(A, c) (C ZS) is a discrete set, each set T(Aj(r(A, c), xo) has non-empty interior, so that it becomes an s dimensional body. Hence, the tessellation R(A, c; j ) consists of polyhedra of dimension s. Now, we suppose c = oo. Choose any xo E Ajr(A, oo) for a given number 0 5 j < oo. Then, by (lg), we get (20) with cm = 0, c = oo. Hence, we can show that T(Ajr(A,oo),xO) is compact as we have shown in the case 1 < c < oo, which completes the proof of the first statement of the theorem. We prove the second statement. Suppose c < oo. We put

I

1 I 1

I I

which is a word over {O, 1) on the set ZS, where ~s is the characteristic map with respect to a set S. If Q(A, c; j ) is C-periodic, then so is the set A j r . Applying ~ - toj the set A j r , we see that the set r is C-periodic. Hence, we get the C-periodicity of A j r for all 0 2 j < c, which implies the C-periodicity of Wj(A; c) for all 0 2 j < c. Then, we can conclude that W(A; c) is C-periodic. Conversely, the C-periodicity of W(A; c) implies that of Wj(A; c) for each 0 6j < c, so that all the sets Ajr(A; c) are C-periodic, and so are the tessellations R(A, c;j), 0 5 j < c. We prove the third assertion. First, we prove the latter half. In view of (18), we see that Aj(ZS) is a *-periodic set for all 0 6j < oo. Hence, (19) together with Aj(ZS) > Ajc'(ZS) implies that for any 0 2 j < oo, the set AjF(A, oo) is a 9-periodic set as a mosaic with periods coming from (18) with j 1 in place of j, so that the tesselation R(A, oo; j ) is *-periodic for any j 2 0. Secondly, we prove the first half. We choose an element gn E An(ZS)\ {o) among elements having the smallest norm in the set An(ZS)\ {o) for each n. Then we can show that

+

Certain words, tilings, their non-periodicity, and substitutions of. . .

339

Suppose the contrary. Then we can choose a bounded subsequence {gp(n)}n=0,1,2,... ( l i m ~ ( n= ) m) of the sequence {gn}n=0,1,2,..: Then we can choose a subsequence {gq(n)}n=o,l,2,... (lim q(n) = oo) of {gp(n))n=0,1,2,... such that it converges to an element g E An(ZS)\ {o). Then A'(")(Z~) 3 gq(,) = g # o holds for all sufficiently large n. Hence, we get A-'(")~ E ZS, g # o for infinitely many n , which contradicts A E (Bdd). Thus, we get (22). Hence, we see that any choice x,, y n E A n r (A, m ) = Tn(A) = An (ZS) \ An+' (ZS) c An (Zs),

namely

F, o) becomes a parallelepiped of dimension s with 2s Thus, T(-x vertices V(E) (E: E (1, -I)'), SO that

-

1

I

I I

I

I

Now, suppose that W = W(A; oo) is C-periodic, i e , Wlx X E C(ZS), t # o. Then we have

= Wlt+~,

where Wn (A; oo) is the word defined by (21). Recalling that Anr(A, oo) is a 9-periodic set as a mosaic with periods coming from (18) with j = n+1, wecan find x n , y n E Anr(A,oo)nX (x, # y,), n = 0,1,2,. . . for any spreading set X E C(ZS). Therefore, we get lit 11 2 llgn 11 by (23), (24). Taking n -+oo, we have a contradiction by (22). We remark that some tilings a ( A , c; j ) for singular matrices A are *-periodic, cf. the examples (i-iii), Section 4. Now, we are intending to show that some of the words W(A; c) and tessellations R(A, c; j ) are non-C-periodic for 1 < c < oo, A E (Bdd).

Lemma 6. Let W be a word on ZS, and U E GL(s; Z). Then W is non*-periodic (resp., non-C-periodic) if and only zf so are the (T, U)-sectors for all T E (1, -1)'. Proof. By our definitions, any (7, U)-sector of W is non-*-periodic (resp., non-C-periodic) if so is W. We assume that all the (T, U)-sector of W are non-*-periodic (resp., non-C-periodic) . It suffices to show that W is non- @-periodic (resp., non-C-periodic) for any X = p + J with p E ZS and J E 9, (resp., X E C,). Suppose the contrary. Then Wlx is *-periodic (resp., Cperiodic) for some X = p J with p E ZS and J E QS (resp., X E Cs). Then J n U(T 8 NS) E Qs n ZS (resp., X n U(T 8 NS) E C(ZS)) holds at least one element T E (1,-1IS. Hence, (7, U)-sector of W is Q-periodic (resp., C-periodic) , which contradicts our assumption.

lx

+

=g 340


Certain words, tilings, their non-periodicity, and substitutions of. . .

341

Apart from the partitions (2), we may consider substitutions of dimension s having fixed points which are non-C-periodic.

Theorem 5. Let a be a substitution over K = {@) U FoU Fx of size G(b) w i t h F o n F x = 4, b = T ( b l , - . . ,bs), bi > l ( 1 5 i 5 s), s > 1, ~~ a ( a ) = ( ~ , ( a ) ) , satisfying vo(@ ) = @ ; vo(a) E Fofor all a E Fo, vo(b) E Fx for all b E Fx; v,(a) E Fofor all a E K , and x E Go := { T ( x l , . . - , x s ) E G(b) \ (0); x1 . . . x s = 01, v,(a) E Fx for all a E K , and x E G x := {T(xl, - . . ,xs) E G(b);

XI-

. . xs # 0).

Then the fied point W E K ~ (W(o) ' = @ ) of a is non-C-periodic.

N.B.: We mean by X I . . xs = 0 not a word but the product of numbers XI,

. . . ,xs equals zero.

Figure 1:

Proof. Let a have the property stated in Theorem 5. Since, for any E (1, -I)', and x E G(b) \ {o),

T

all the conjugates of have the same property. We consider the automata rM = (@ , K , G(b), rC) with TC corresponding to r a . Then, all the conjugates rM can be described as in Fig.1 given below. We mean the transition function T( by the arrows labeled by Go, G x , or o there. For instance, for a E Fx,' 1 of the field Q ( @ ) . An irrationality measure for the number is obtained by Bombieri's method. See Section 4. After an extensive work done by Bennett and B. M. M. de Weger [BedW], recently Bennett [Be21 proved, by using both the Pad6 approximation method and Baker's method, that for d 3 the equation axd - byd = 1 has at most one positive solution. This can be considered to be a final result about family (iv) with k = 1. There are some other families of Thue equations of degree > 3 which were treated by the Pad6 approximation method.

+

>

>

+

(v) 1x4 - ax3y - 6x2y2 axy3 + y41 5 k. Chen Jianhua and P. M. Voutier [CheV] solved this family with k = 1 for a 128. They use an important lemma of Thue [Thu3] which provides a possibility to apply the Pad6 approximation method to some 3. For general k, Lettl, special type of Thue equations of degree Petho, and Voutier [LPV] gave an upper bound for the solutions in the case a 58, and proved that, in the case k = 6a 7 and a 58, the only solutions with 1x1 5 y are (0,1), (&I, 1))(&I,2). Historically, before [CheV],by using Baker's method Lettl and Petho [LP] had already solved completely the corresponding family of Thue equations (not inequalities) for k = 1 and 4.

>

>

>

+

>

(vi) ~x6-2ax5y-(5a+15)x4y2-20x3y3+5ax2Y4+(2a+6)xy5+y6~ 5 k. Lettl, Petho, and Voutier [LPV] gave an upper bound for the solutions in the case a 89, and solved the inequality completely in the case k = 120a+323 and a 2 89. Families (i), (v) and (vi) are called simple families since the solutions of the corresponding algebraic equation F ( x , 1) =

>

365

0 are permuted transitively by a fractional linear transformation with rational coefficients (see [LPV] for the definition). (vii) . . .1x4 - a2x2y2- by4[ 5 k. For b = -1, the author [Wakl] gave an upper bound for the solutions in the case a 8, and solved the inequality when k = a 2 - 2 and a 8. When b 1 and a is greater than a certain value depending on b, an upper bound for the solutions was obtained by [Wak2], and the case b = 1,2, a 2 1, and k = a2 b - 1 was completely solved. The method is based on Pad6 approximations. In the Pad6 approximation method, one usually constructs, for a real solution 8 of the corresponding algebraic equation, linear forms in 1,8. In these two works however, linear forms in 1,8, e2 were used, and it seems that linear forms of this kind were used for the first time to solve Thue equations.

>

> >

+

4.

BOMBIERI'S METHOD

The proof of the finiteness theorem of Thue on equation (1) is based on the idea that if there is a rational number which is sufficiently close to a given real number 8, then other rational numbers can not be too close to 8. In fact, Thue first assumes existence of a rational number polgo having sufficiently large denominator and exceptionally close to 8. Then, taking another plq close to 8, and using the box principle, he constructs an auxiliary polynomial P(x, y) which vanishes at (8,8) to a high order. And he proves that P vanishes at (po/qo,p/q) only to a low order. To prove this, he needs the assumption that qo is sufficiently large. However, this assumption for polgo is too strong, and no pair (8,po/q0) having the required property is known at present, and by this reason no effective result has been found using this method. E. Bombieri [Boll however succeeded, by using Dyson's lemma, to remove the requirement that qo should be large. This lemma asserts in fact that P vanishes at (po/qo,p/q) only to a low order, by using information on the degree of P and the vanishing degree at (B,O) only. Hence it is free from the size of qo, and it allows to use an approximation po/qo to 8 even if its denominator is small. Thus, Bombieri could find examples of good pairs (O,po/qo), and obtained effective irrationality measures for certain algebraic numbers. Example [Boll. Let d 2 40, and let B be the positive root of xd-axd-'+I, where a 2 A(d) and A(d) is effectively computable. Then, O has effective irrationality measure X = 39.2574.

366

On families of cubic Thue equations


Refining the above method, Bombieri and J. Mueller [BoM] obtained the following.

Theorem ([BoM]). Let d

> 3,

and let a and b be coprime positive

$(m)

integers, and put p = log l a - b ' . Let 6' E be of degree d . If log b p < 1 - 2 / d , then for any E > 0, 0 has eflective irrationality measure A=-

2

d5 log d

1-P Pursuing this direction, Bombieri [Bo2] in 1993 succeeded to create a new method for obtaining an effective irrationality measure for any algebraic number. This method applies to all algebraic numbers, hence it is a very general theorem as well as Baker's theorem. The method is completely different from that of Baker. Bombieri, A. J. van der Poorten, and J. D. Vaaler [BvPV] applied Bombieri's method to algebraic numbers of degree three over an algebraic number field, and under a certain assumption, obtained an irrationality measure with respect to the ground field. For simplicity, we state their result for cubic numbers over the rational number field.

Theorem ([BvPV]). Let a (# 0) and b be integers (positive or negative), and f ( x ) = x3 a x b be irreducible. Let 6' be the real root of f whose absolute value is the smallest among the three roots of f . Assume that la1 > eloo0 and la1 b2. Then, 6' has eflective irrationality measure

+ + >

+

367

gives very precise results on the maximal number of solutions, and it also gives an alternative proof of Thue's finiteness theorem for these special cases. Nagell [N4], and Delone and Faddeev [DF] are good books on this subject. Families (viii) and (ix) are special cases of (iv). (viii) ax3 + by3 = k . Case b = 1 and k = 1. Delaunay and Nagell independently proved that this equation has at most one solution with x y # 0. For the proof, see [Dell, [Nl] and [N2]. Case k = 1 or 3. Nagell [Nl] [N2] proved that this equation has at most one solution with xy # 0 except the equation 2x3 y3 = 3 which has two such solutions ( 1 , l ) and (4, -5).

+

(ix) x4 - ay4 = 1. Tartakovskij [Ta] proved that if a # 15 then this equation has at most one positive solution. Actually, one can see by using KANT that the same holds for a = 15 also. (x) F ( x , y ) = 1 where F is a homogeneous cubic polynomial with negative discriminant. The assumption that the discriminant is negative implies that F ( x , 1) has only one real zero and the associated unit group has rank 1. Delaunay [De2] and Nagell [N3] independently proved that, if the discriminant is not equal t o -23, -31, -44, then this equation has at most 3 solutions. (xi) x 3 + a x Y 2 + y 3 = 1, a > 2 . The only solutions of this equation are (1,O), ( 0 , l ) and (1,-a). This is a corollary of the above general result of Delaunay and Nagell on equation (x) since the discriminant -4a3 - 27 is negative by the assumption and the equation has already three solutions. The case a = 1 corresponds to one of the exceptional cases.

Example. For la1 = [eloo0] 1 and lbl = 1 we have X = 2.971. Remark. If a > 0, then f has only one real root, and its absolute value is smaller than that of the complex roots of f .

5.

ALGEBRAIC METHOD

After Thue proved his theorem on the finiteness of the number of solutions, the maximal number of solutions for some special families of Thue equations of low degrees was determined by the algebraic method. This method uses intensively properties of the units of the associated number field or of the associated order. Therefore, only the cases where the units group has rank 1 have been treated. However, this method

6.

PRINCIPLE OF THE PADE APPROXIMATION METHOD

We explain here how we apply Pad6 approximations to solve Thue equations. For this purpose, let us consider equation (4), namely the example treated by Thue and Siegel. The principle is as follows. In order to solve this equation, we need to obtain properties concerning rational approximations to the algebraic number a = with d = 3. Let us note that this number is written as a = In order to obtain properties of this number, we first consider the binomial function and obtain some property concerning Pad6 approximations to

q

w

q m .

vm,

A

368

ANALYTIC NUMBER T H E O R Y

this function. Then, putting x = l l a and using the property of this function, we obtain properties of the number a. In order to explain what Pad6 approximations are, we state the following proposition.

Proposition. Let f (x) be a Taylor series at the origin with rational coeficients. Then, for every positive integer n, there exist polynomials Pn(x) and Qn(x) (Q, # 0) with rational coeficients and of degree at most n such that

holds, namely, such that the Taylor expansion of the left-hand side begins with the term of degree at least 2 n + 1. We call the rational functions Pn(x)/Qn (x) or the pairs (Pn(x), Qn (x)) Pad6 approximations to f (x) .

Proof. The necessary condition for Pn and Qn is written as a system of linear equations in their coefficients. Comparing the number of equations and the number of unknowns, we find a non-trivial solution. By this proposition we see that Pad6 approximations to a given function exist always. For application to Diophantine equations, this existence theorem is not sufficient, and it is very important to know properties of the polynomials Pn(x) and Qn(x). Namely, it is necessary to know the size of the denominators of their coefficients, and upper bounds for values of these polynomials and the right-hand side. Therefore, it is important to be able to construct Pad6 approximations concretely.

Example (Thue-Siegel). Pad6 approximations to the binomial function $ C F Z are given by

where F denotes the hypergeometric function of Gauss (in this case they are hypergeometric polynomials). We put x = l / a into this formula, and we multiply the relation by the common denominators of the coefficients of these hypergeometric polynomials and also by a n . Then we obtain


t

369

Suppose a is sufficiently large. Then we can verify that if n tends to the infinity, then the right-hand side tends to zero. From this we obtain a sequence of rational numbers pn/qn which approach to the number a= Since we know well about hypergeometric functions, we can obtain necessary information about p,, q,, and the right-hand side of the above relation. Then, comparing a rational number p/q close to the number a with the sequence pn/qn, we obtain a result on rational approximations to a. (Actually we should construct two sequences approaching to a . ) This final process is based on the following lemma. Its idea is due to Thue.

q m .

Lemma. Let 8 be a non-zero real number. Suppose there are positive numbers p, P, I , L with L > 1, and further there are, for each integer n 2 1, two linear forms

with integer coeficients pin and gin satisfying the following conditions: (i) the two linear forms are linearly independent; (ii) [gin1 p P n ; and (iii) lltnl l / L n . Then 8 has irrationality measure

<
0. We give new results on the family of

7.

cubic Thue inequalities given by (3). Our method is based on Pad6 approximations. We give an outline of the proof in Section 8. The full proof will appear elsewhere. As mentioned above, there are preceding results concerning this family: Equation (xi) was solved by algebraic method; irrationality measures for the associated algebraic numbers were given by Bombieri's method; Chudnovsky gave irrationality measures for the case b = 1 and a -3(mod 9). (In the last two cases, the results hold for a < 0 also.) P u t f (x) = x3+ax+ b. From the assumption that a > 0, the algebraic equation f (x) = 0 has only one real solution and the other two solutions are complex. Let us denote by 8 this real solution. Further we put

-

Theorem 1. Suppose a > 22/334b8/3(1

. Then for any

213

integers p, q (q > O), we have

X(a, b) = 1

3

+ 3g01b13

+ log((1log(4dZ + 12&1b1) < 3. - 27b2/~)dZ/(27b2))

Further, X(a, b) is a decreasing function of a and tends to 2 when a tends to 00. Remark. The above assumption on the size of a is imposed in order to obtain the inequality X(a, b) < 3 which is the essential requirement for application. For example, we obtain X(a, b) < 3 for b = 1 and a 2 129, and for b = 2 and a 2 817. This shows that for small b we have X(a, b) < 3 for relatively small a. Compare with the assumption in the theorem of Bombieri, van der Poorten, and Vaaler mentioned in Section 4. Moreover, in their result, X behaves asymptotically for large a

while ours behaves X(a, b)

-

2

+ 2 log 108 + 4 log 1b13 log a ,

and behaves better. However we should note that their result holds for cubic algebraic numbers over any algebraic number field. Compare also with the behavior of X in Chudnovsky's theorem. As an easy consequence of Theorem 1, we obtain the following.

Theorem 2. Under the same assumption as in Theorem 1, we have, for any solution (x, y ) of the Thue inequality (3),

with the same X(a, b) as in Theorem 1. Since we have obtained an upper bound for the solutions of (3), we may consider that (3) is solved in a sense under the assumption. In order

372



to find all solutions of ( 3 ) completely, we need to specify the value k. Let b > 0. Taking into account that for (x, y ) = ( 1 , l ) the left-hand side of ( 3 ) is equal to a b 1, let us put k = a b 1, and let us consider the Thue inequalities

+ +

+ +

Theorem 3. Let b > 0 and a 2 3000b4. Then the only solutions of (5) with y 2 0 and gcd(x, y ) = 1 are

Let us call these solutions the trivial solutions of (5) with y For b = 1,2 we can solve (5) completely.

2 0.

>

Theorem 4. Let b = 1 or b = 2, and let a 1. Then the only solutions of (5) with y 2 0 and gcd(x, y ) = 1 are the trivial solutions except the cases b = 1, 1 5 a 5 3 and b = 2, 1 5 a 5 7. Further, we can list up all solutions for the exceptional cases.

373

In order to obtain the form of the formula, we just set a = s + tJfi and ?!, = u + v f i with unknowns s , t, u , v, and put them into the righthand side, and we compare the coefficients. Then we obtain the formula. This fact was used also in Siege1 [Sl] to determine the number of solutions of cubic Thue equations, and also it was used in Chudnovsky [Chu]. This made possible to obtain results on general cubic numbers of wide class by considering only the binomial function and its Pad6 approximations. We use this formula in a different way from theirs. Since 8 is a real zero of f (x), we have

The key point of our method is to consider the number on the left-hand side, and to apply to this number Pad6 approximations to the function V ( 1 - x)/(l x ) . To construct Pad6 approximations, we use Rickert's integrals [R].

+

)L

8.

SKETCH OF THE PROOF

Proof of Theorem 1. We use the Pad6 approximation method expiained. in Section 6. As explained above, in order to solve equation (4) or the more general equation (iv), it was necessary to obtain a result on rational approximations t o the associated algebraic number. Since those equations have diagonal form, namely they contain only the terms xd and y d , the associated algebraic number is a root of a rational number. Therefore the binomial function was used. However, in our case, equation (3) has not diagonal form. But this inconvenience can be overcome by transforming the equation into an equation of diagonal form as follows. The transformation itself is an easy consequence of the syzygy theorem for cubic forms in invariant theory, namely the relation 47-t3 = D F 2 - 92among a cubic form F, its discriminant D, and its covariants 7-t and 9 of degree 2 and 3 respectively. Lemma 1. The polynomial f can be written as

Lemma 2. For n 2 1, i = 1,2 and small x, let

where Ti ( i = 1,2) is a small simple closed counter-clockwise curve enclosing the point 1 (resp. -1). Then these integrals are written as

where Pn(x) is a polynomial of degree at most n with rational coefficients. Further we have

Now we put s = 3 f i b / J ? i into these Pad6 approximations. Then using (6) we rewrite the left-hand side in terms of 8, P and p, and we multiply by 8 + Then we observe that the left-hand side does not contain any square root. Thus we obtain linear forms

p.

where

Proof. We can verify the formula by a simple calculation starting from the right-hand side.

with rational coefficients. Since the Pad6 approximations are given by integrals, we can obtain necessary information about the linear forms by residue calculus and by



B. N. Delone and D. K. Faddeev, The Theory of Irrationalities of the Third Degree, Translations of Math. Monographs, Vol. 10, Amer. Math. Soc., 1964. J.-H. Evertse, On the equation axn- byn = c, Compositio Math., 47 (1982), 289-315.

P

[N3] [N4]

[PI

,

On the representation of integers by binary cubic forms of positive discriminant, Invent. Math., 73 (l983), 117-138.

C. Heuberger, On general families of parametrized Thue equations, in "Algebraic Number Theory and Diophantine Analysis", F. Halter-Koch and R. F. Tichy, Eds., Proc. Conf. Graz, 1988, W. de Gruyter, Berlin (2000), 215-238. C. Heuberger, A. Petho, and R. F. Tichy, Complete solution of parametrized Thue equations, Acta Math. Inform. Univ. Ostraviensis, 6 (1998), 93-1 13.

M. Laurent, M. Mignotte, and Y. Nesterenko, Formes linkaires en deux logarithmes et dkterminants d'interpolation, J. Number Theory, 55 (l995), 285-321. G. Lettl and A. Petho, Complete solution of a family of quartic Thue equations, Abh. Math. Sem. Univ. Hamburg, 65 (1995), 365-383. G. Lettl, A. Petho, and P. Voutier, Simple families of Thue inequalities, Trans. Amer. Math. Soc., 351 (1999), 1871-1894. F. Lippok, On the representation of 1 by binary cubic forms of positive discriminant, J. Symbolic Computation, 15 (1993), 297-313. M. Mignotte, Verification of a conjecture of E. Thomas, J. Number Theory, 44 (1993), 172-177. , Petho's cubics, Publ. Math. Debrecen, 56 (2000), 481505. M. Mignotte, A. Petho, and F. Lemmermeyer, On the family of Thue equations x3 - (n - 1 ) - ( n~ + 2)xy2 ~ ~ y3 = k, Acta Arith., 76 (1996), 245-269. M. Mignotte and N. Tzanakis, On a family of cubics, J. Number Theory, 39 (1991), 41-49. T . Nagell, Solution complkte de quelques kquations cubiques & deux indkterminkes, J. Math. Pures. Appl., 6 (1925), 209-270. , Uber einige kubische Gleichungen mit zwei Unbestimmten, Math. Z., 24 (1926), 422-447.

[R]

[Sl]

[S2] [Ta] [Thol] [Tho21 [Thul]

[Thu2] [Thu3]

[Thu4]

[Wakl] [WakS] [Wall

,

377

Darstellung ganzer Zahlen durch binare kubische Formen mit negativer Diskriminante, Math. Z., 28 (1928), 10-29. , L'analyse inde'termine'e de dergre' supe'rieur, Memorial Sci. Math. Vol. 39, Gauthier-Villars, Paris, 1929. A. Petho, On the representation of 1 by binary cubic forms with positive discriminant, in "Number Theory, Ulm 1987 Proceedings," H. P. Schlickewei and E. Wirsing, Eds., Lecture Notes Math. Vol. 1380, Springer (1989), 185-196. J. H. Rickert, Simultaneous rational approximations and related Diophantine equations, Math. Proc. Cambridge Philos. Soc., 113 (1993), 461-472. C. L. Siegel, ~ b e einige r Anwendungen diophantischer Approximationen, Abh. Preuss. Akad. Wiss. Phys. Math. K1. No.1, (1929) ; Gesam. Abh. I, 209-266. , Die Gleichung axn - byn = c, Math. Ann., 114 (1937), 57-68 ; Gesam. Abh. 11, 8-19. V. Tartakovskij, Auflosung der Gleichung x4 - py4 = 1, Bull. Acad. Sci. de I'URSS, 1926, 301-324. E. Thomas, Complete solutions to a family of cubic Diophantine equations, J. Number Theory, 34 (IggO), 235-250. -, Solutions to certain families of Thue equations, J. Number Theory, 43 (l993), 319-369. A. Thue, Bemerkungen iiber gewisse Naherungsbriiche algebraischer Zahlen, Kristiania Vidensk. Selsk. Skrifter., I, Mat. Nat. Kl., 1908, No.3. , ~ b e Annaherungswerte r algebraischer Zahlen, J. Reine Angew. Math., 135 (IgOg), 284-305. , Ein Fundamentaltheorem zur Bestimmung von Annahe-rungswerten aller Wurzeln gewisser ganzer Funktionen, J. Reine Angew. Math., 138 (1910), 96-108. , Berechnung aller Losungen Gewisser Gleichungen von der Form axT - byT = f , Kristiania Vidensk. Selsk. Skrifter., I, Mat. Nat. Kl., 1918, No.4. I. Wakabayashi, On a family of quartic Thue inequalities I, J. Number Theory, 66 (1997)) 70-84. , On a family of quartic Thue inequalities 11, J. Number Theory, 80 (2000), 60-88. M. Waldschmidt, Minorations de combinaisons linkaires de logarithmes de nombres alg6briques, Canad. J. Math., 45 (1993),

TWO EXAMPLES OF ZETA-REGULARIZATION Masami YOSHIMOTO Research Institute for Mathematical Sciences, Kyoto University Kyoto, 606-8502, Japan [email protected]

Keywords: zeta-regularization, Kronecker limit formula Abstract

In this paper we shall exhibit the close mutual interaction between zetaregularization theory and number theory by establishing two examples; the first gives the unified Kronecker limit formula, the main feature being that stated in terms of zeta-regularization, the second limit formula is informative enough to entail the first limit formula, and the second example gives a generalization of a series involving the Hurwitz zetafunction, which may have applications in zeta-regularization theory.

1991 Mathematical Subject Classification: 1lM36, 1lMO6, 1lM41, 1lF20

1.

INTRODUCTION

In this paper we shall give two theorems, Theorems 1 and 2, which have their genesis in zeta-regularization. One (Theorem 1) is the second limit formula [16] of Kronecker in which effective use is made of the zetaregularization technique, and the other (Theorem 2) is a generalization of a formula of Erdklyi [6] (in the setting of Hj. Mellin [13]) which gives a generalized zeta regularized sequence (Elizalde et a1 [4]). First, we shall give the definition of the zeta-regularization. Definition. Let {Ak}k=l,l,.,. be a sequence of complex numbers such that 03

for some positive 0. Ah-' for s such that Re s = o Define Z(s) by Z(s) = call it the zeta-functionassociated to the sequence {Ak}. 379 C. - -Jia - - -and --- K. Matsumoto (eds.), Analytic Number Theory, 379-393. . . . - . .. . - . , . , , ., , ,

.

> a and

380


Two examples of zeta-regularization

We suppose that Z(s) can be continued analytically to a region containing the origin. Then we define the regularized product, denoted by A,, of {Ak}

n k

as

k

= log

formula in the form

z

where FP f (so) indicates the constant term in the Laurent expansion of f (s) at s = so. If this definition is meaningful, the sequence {Ak} is called zeta-regularizable. Similarly, if Z(s) can be continued analytically to a region containing s = -1, the zeta-regularized sum is defined by

Cz

381

{ vZ

where the prime "I"indicates that the pair (m, n ) = (0,O) should be excluded from the product, q(z) denotes the Dedekind eta-function defined by 00

On the other hand, Quine et a1 [15], Formula (53), states Kronecker's second limit formula in the form

nZcy +

W) = iq-l(z) exp

exp(hk)} = FP Z(- 1).

Giving a meaning in this way to the otherwise divergent series or products by interpreting them as special values of suitable zeta-functions (or derivatives thereof) is called Zeta-regularizat ion.

nE1

We shall now state the background of our results more precisely. Regarding Kronecker's limit formulas, J. R. Quine et al [15] were the first who used the zeta regularized products to derive Kronecker's first limit

- Tiw) B ~ ( wz), ,

(3)

where y runs through all elements m + n r of the lattice with basis 1, T (-T 5 arg y < T) and &(w, z) is one of Jacobi's theta-function given by 00

B1(w,z) =

Remark 1. i) As {Ak) we consider mostly the discrete spectrum of a differential operator. ii) Formal differentiation gives

so that this gives a motivation for interpreting the formal infinite product Xk as e-FP "(O). iiij The merit of the zeta-regularization method lies in the fact that by only formal calculation one can get the expressions wanted, save for the main term, which is given as a residual function (the sum of the residues), for more details, see Bochner [3]. To calculate the residual function one has to appeal to classical methods. Cf. Remark 2 and Remark 3, below.

(-T

C exp n=-00

Quine et a1 [15] proved (3) using a generalization of Voros' theorem [19] on the ratio of the zeta-regularized product and the Weierstrass product to the case of zeta regularizable sequences (Theorem 2 [15]). It is rather surprising that they reached (3) without knowing the first form of Kronecker's second limit formula (from the references they gave they do not seem to know Siegel's most famous book [16] on Kronecker limit formulas) as (3) does not immediately lead to it (the conjugate decomposition property does not necessarily hold for complex sequences

4-

Siegel's book [16] had so much impact on the development of Kronecker's limit formulas and their applications in algebraic number theory that the general understanding before Berndt's paper [I] was that the first and the second limit formulas of Kronecker should be treated separately, and may not be unified into one (see e.g. Lang [ll]and others). Berndt [I] was the first who unified these two limit formulas into one, but this paper does not seem to be well-known for its importance probably because it was published under a rather general title "Identities involving the coefficients of a class of Dirichlet series".

382

Two examples of zeta-regularization


It is also possible to unify the investigation of Chowla and Selberg with Siegel to deduce a unified version of Kronecker's limit formula (Kumagai [lo]), but this is superseded by Berndt's theorem in the sense that in Berndt's one is to take only w = 0 while in Kumagai's paper, the residual function has not been extracted out. To restore the importance of Berndt's paper and clarify the situation surrounding Kronecker's limit formulas we shall prove the unified Kronecker limit formula. The main feature is that, stated in terms of zeta-regularized product, the second limit formula is informative enough t o entail the first limit formula:

383

Theorem 1. i) cQ(s;u, v) has the expansion at s = 1, tI

where and

1

g1(U- UZ, 2) - (1 - E(U, u)) log lu - u z } - log (u - uz>rl(z>

,

(10)

where E(U, u) is defined by Indeed, noting that &(u,21) = we can take the limit in (5) as w

-+

0, and

1, (u, 4 = (090) 0, otherwise.

ii) Under the convention (1 - E(u,u))log Iu - uzl = 0 for (u,u) = (0,0), we can rewrite (8) as Kronecker's first limit formula stated in Siegel [16].

To state Theorem 1 we fix the following notation. Let Q((, q) = ac2 2btq q2 be a positive definite quadratic form with a > 0, and discriminant d = ac - b2 > 0 which we decompose as

+

+

where y is Euler constant and ((s) = C(s, 1) = Riemann zeta-function.

xzln-'

denotes the

We now turn to the second example, which is discussed in Elizalde et a1 [4] under the title "Zeta-regularization generalized". Our main object of study is the function

> It1

391

(29)

in view of (19). Divide the integration path of RN(s,T) into three parts: v = Im t E (-oo,-T), [-T,T), [T,co) and denote the corresponding

where cl ( a ) = c(a){alog 2n - a log a,

+ (1 - a ) (log c(a) - 1)).

392

TWOexamples of zeta-regularization


Thus, from (31)) (32) and (33), it follows that

with c2(a) = cl(a), c;(a) = a ( l o g 2 ~ loga Hence, as long as 0 < a < 1, we have

a.N

+

oo and for a = 1, c;(l) = log 2~

+ 1) - 1.

> 0, we have

asN-+oo. By (35) and (35)' we conclude (27). This completes the proof.

0

Remark 3. We transform formally F(s,T) by the method of zeta-regularization to obtain

00

= k=O

H C ( a ( s- k), a ) k!

+ (a correction term)

with s - # O,1,2,. . . In fact P(s,T) is the correction term to be calculated by another met hod.

References [I] B. C. Berndt, Identities involving the coeficients of a class of Dirichlet series. VI, Trans. Amer. Math. Soc. 160 (1971)) 157-167. [2] B. C. Berndt, Ramanujan's notebooks. Part I, Springer-Verlag, New York-Berlin, 1985. [3] S. Bochner, Some properties of modular relations, Ann. of Math. (2) 53 (1951)) 332-363. [4] E. Elizalde, et al, Zeta regularization techniques with applications, World Scientific Publishing Co. Pte. Ltd., 1994. [5] P. Epstein, Zur Theorie allgemeiner Zetafunctionen, Math. Ann. 56 (1903), 615-644. [6] A. ErdBlyi, et al, Higher transcendental functions, McGraw-Hill, 1953.

393

[7] L. Euler, Introduction to the analysis of the infinite, Springer Verlag, 1988. [8] M. Katsurada, On Mellin-Barnes type of integrals and sums associated with the Riemann zeta-function, Publ. Inst. Math. (Beograd) (N.S.) 62(76)(1997)) 13-25. [9] H. Kumagai, The determinant of the Laplacian on the n-sphere, Acta Arith. 91 (lggg), 199-208. [lo] H. Kumagai, On unified Kronecker limit formula, Kyushu J. Math., to appear. [l 11 S. Lang, Elliptic functions, Addison-Wesley, 1973. [12] M. Lerch, Dalii studie v oboru Malmste'novskych Tad, Rozpravy ~ e s k 6Akad. 3 (1894) no.28, 1-61. [13] Hj. Mellin, Die Dirichletschen Reihen, die zahlentheoretischen Funktionen und die unendlichen Produkte von endlichem Geschlecht, Acta Soc. Sci. F e n n i c ~31 (1902), 3-48. [14] T . Orloff, Another proof of the Kronecker limit formulae, Number theory (Montreal, Que., 1985), 273-278, CMS Conf. Proc., 7, Amer. Math. Soc., Providence, R. I., 1987. [15] J. R. Quine, S. H. Heydari and R. Y. Song, Zeta regularized products, Trans. Amer. Math. Soc. 338 (1993), 213-231. [16] C. L. Siegel, Lectures on advanced analytic number theory, Tata Inst. 1961, 2nd ed., 1980. [17] R. Song, Properties of zeta regularized products, Thesis, Florida State University, 1993. [18] H. M. Stark, L-Functions at s = 1. 11,Advances in Math. 17 (1975), 60-92. [19] A. Voros, Spectral functions, special functions and the Selberg zeta function, Comm. Math. Phys. 110 (l987), 439-465.

A HYBRID MEAN VALUE FORMULA OF DEDEKIND SUMS AND HURWITZ ZETA-FUNCTIONS ZHANG Wenpeng Department of Mathematics, Northwest University, Xi'an, Shaanxi, P. R. China

Keywords: Dedekind sums, Hurwitz zeta-function, The hybrid mean value Abstract

The main purpose of this paper is using the mean value theorem of the Dirichlet L-functions and the estimates for character sums to study the mean value distribution of Dedekind sums with a weight of Hurwitz zeta-function, and give an interesting asymptotic formula.

2000 Mathematics Subject Classification: llN37.

1.

INTRODUCTION

For a positive integer k and an arbitrary integer h, the Dedekind sum S ( h , k ) is defined by

where x - [XI -

4

if x is not an integer; if x is an integer.

Various properties of S ( h , k ) were investigated by many authors. For example, Carlitz [3], Mordell [5] and Rademacher [6] obtained an important reciprocity formula for S ( h , k ) . Conrey et al. [4] studied the mean value distribution of S ( h , k ) , and gave an interesting asymptotic formula. The main purpose of this paper is to study the asymptotic

This work is supported by the P.N.S.F. and N.S.F. of P. R. China.

395 C. Jia and K. Matsumoto (eds.),Analytic Number Theory, 395408.

396

A hybrid m e a n value formula of Dedekind S u m s . . .


properties of the hybrid mean value

where ((s, a ) is Hurwitz zeta-function,

397

Corollary 2. Let p be an odd prime, A = A(p) denotes the set of all quadratic residues modulo p i n the interval [I,p-11. Then the asymptotic formula X I

denotes the summation over

a

all integers a coprime to q. Regarding (I), it seems that it has not yet been studied, at least I have not seen expressions like (1) before. The problem is interesting because it can help us to find some new relationship between Dedekind sums and Hurwitz zeta-functions. In this paper, we shall give a sharper asymptotic formula for (1). The constants implied by the 0-symbols and the symbols 3, ( a ,q ) = 1.

Then we have

d2 Y mod d

where +(d) is the Euler function, x denotes a Dirichlet character modulo d with x(-1) = -1, and L ( s , x ) denotes the Dirichlet L-function corresponding to X.

where L ( s , X ) is Dirichlet L-function and exp(y) = e'. From this Theorem we may immediately deduce the following two Corollaries:

Proof. (See reference [9]).

Corollary 1. Let q be an integer with q 2 3. Then for any fixed positive integer m and n, we have the asymptotic formula

Lemma 2. Let q be any integer with q >_ 3, x denotes an odd Dirichlet character modulo d with dlq, X I be any Dirichlet character modulo q. Then for any fixed positive integer m, we have the asymptotic formula

x mod d x(-I)=-1 where

C' denotes the summation over all a such that ( q , a ) = 1, n a

PI*

denotes the product over all prime divisors of q, and C(s) is the Riemann zeta-function.

Proof. For the sake of simplicity we only prove that Lemma 2 holds for m = 1. Similarly we can deduce other cases. For any character

398


xq modulo

A hybrid mean value formula of Dedekind Sums . . .

399

x q ( a ) , and X: denote the principal

q , let A ( y , x q ) = d 0, applying Abel's identity we have

xq(# Xi)modulo q and

In-m( mod d ) ln>d

W)

--

2

c

1 j l < d l < m < d l

Analytic Number Theory

Analytic number theory

Analytic number theory